Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 5, No. 3, 2023 

 

79 

Optimization	Model	of	Survey	Line	Layout	for	
Multibeam	Bathymetric	Systems	
Shenrui Lin †, Yuangang Li †, Yaru Wang † 

School of Computer Science and Engineering, Northeastern University, Shenyang, China 
† These authors also contributed equally to this work 

 

Abstract: Multibeam bathymetric system is an underwater terrain measurement technology with high efficiency, high accuracy 
and high resolution. Based on the multibeam bathymetric system, this paper establishes an ideal geometric model and an isobath 
iterative model. Through the physical method of ideal model, the actual physical problem of multibeam sounding is abstracted 
into a geometric model about a conic curve, and under the constraint of overlap rate, the coordinates of the next survey line are 
iterated according to the current depth to solve the optimal set of survey lines. First, the seawater depths, strip coverage widths, 
and overlap rates with the previous line are calculated for lines at different distances from the center point. Then, the angle 
between the direction of the survey line and the projection of the normal direction of the seafloor slope on the horizontal plane 
is introduced to solve for the current coverage width of the survey line. Again, the optimal set of survey lines is solved under the 
constraint of the overlap rate. Finally, Sobel edge detection is used to divide the surface of the seafloor terrain into multiple 
planes with defined slope angles to solve the optimal line set for the entire seafloor surface. The structure of the proposed 
mathematical model is clear and simple, which is in line with the actual situation. 

Keywords: Ideal Geometric Model; Conic Curves; Isometric Iterative Model; Sobel Edge Detection; Surface Fitting. 
 

1. Introduction 
The multibeam bathymetric system has a wide range of 

application prospects in China's ocean exploration and marine 
resource management, maintains China's marine environment 
and homeland security, and provides a powerful tool for 
scientific research and ocean management [1-3]. 

Multibeam bathymetric systems utilize multiple acoustic 
beams to simultaneously transmit and receive acoustic signals 
to measure the depth and topography of the water column [4-
5]. Compared with single-beam bathymetry, multibeam 
systems can realize full-coverage bathymetry in flat areas of 
the seafloor, which improves data density and topographic 
information acquisition. 

In order to solve the above problems, this paper establishes 
an ideal geometric model and an isobath iterative model based 
on the multibeam bathymetry system. Through the physical 
method of ideal model, the actual physical problem of 
multibeam sounding is abstracted into a geometrical model 
about a conic curve, and under the constraint of overlap rate, 
the coordinates of the next survey line are iterated according 
to the current depth, thus solving the optimal set of survey 
lines. First, the seawater depths, strip coverage widths, and 
overlap rates with the previous line are calculated for lines at 
different distances from the center point. Then, the angle 
between the direction of the survey line and the projection of 
the normal direction of the seafloor slope on the horizontal 
plane is introduced to solve for the current coverage width of 
the survey line. Again, the optimal set of survey lines is solved 
under the constraint of the overlap rate. Finally, the surface of 
the seafloor terrain is segmented into multiple planes with 
defined slope angles using Sobel edge detection to solve for 
the optimal set of survey lines for the entire seafloor surface. 

2. Model Formulation and Solving 

2.1. Modeling and Solving When the Survey 
Line is Parallel to the Seafloor Slope 

A geometric analysis of the multibeam bathymetry scene is 
shown in Figure 1. 

 
Figure 1. Geometric analysis of the multibeam bathymetry scene 

 
When the survey lines are parallel to each other and the 

seafloor topography is flat, then the overlap rate η between 
neighboring strips is defined as: 

W

d
WW

W

d 
 221               (1) 

Thus defining the overlap rate η between neighboring strips 
on a slope with a gradient α is defined as: 



 

80 

21

031

WW

WWW




                (2) 

That is, the width of the overlap as a percentage of the 
width of the coverage of the line at the shallower depths of 
the sea. 

Let the angle between the left beam and the seafloor slope 
in Fig. 1 be θ1, and the angle between the right beam and the 
seafloor slope be θ2. 

By geometric analysis θ1 and θ2 are satisfied: 


















22

22

2

1

              (3) 

From the sine theorem, the relationship between the 
opening angle θ of the multibeam transducer, the depth D of 
the slope, the angle θ between the right beam and the seafloor 
slope2 and W1 can be expressed as follows: 

1
1

sin
2

sin




nD

W                  (4) 

Likewise: 
























2

1

3

2
2

0

sin
2

sin

sin
2

sin

cos









n

n

D
W

D
W

d
W

             (5) 

From the geometric relationship 
tan1 dDD nn                (6) 

x represents the distance of the survey line from the center 
point, and the horizontal direction pointing to the rise of the 
seabed slope is the positive direction of x. From the geometric 
relationship the coverage width W can be expressed as: 

cos21 ）（ WWW             (7) 
From the geometric model the depth of seawater D can be 

expressed as: 
tandhD C                  (8) 

The opening angle θ of the multibeam transducer is 120°, 
the slope α is 1.5°, the depth of seawater hC at the center point 
of the sea is 70 m, and the spacing of the survey lines d is 200 
m, i.e: 
















md

mhC

200

70
5.1

120




                  (9) 

The spacing of the survey lines d is 200 m, and the distance 
from the survey line to the center point is -800 m. The 
seawater depths, coverage widths, and overlap rates with the 
previous survey line corresponding to different distances of 
the survey lines are obtained as shown in Figure 2. 

 
Figure 2. Plot of data corresponding to measuring lines of different 

distances 
 

From Figure 2, it can be seen that the depth of the sea water, 
the coverage width and the overlap rate with the previous 
measurement line decrease with the increase of the distance 
from the center point, and the overlap rate η is negative at 
x=600m and x=800m, which means that the "leakage 
phenomenon" has occurred. 

2.2. Modeling and Solving Multibeam 
Bathymetric Coverage Width at Different 
Points 

The angle β between the direction of the survey line and 
the projection of the normal direction of the seafloor slope on 
the horizontal plane, the direction of the survey line is 
perpendicular to the beam sector, and the range of β is [0,2
 ], as shown in Figure 3. 

 
Figure 3. Schematic diagram of the angle β between the direction 
of the survey line and the projection of the normal direction of the 

seabed slope on the horizontal plane 
 

The angle β between the direction of the survey line and 
the normal direction of the seafloor slope projected on the 
horizontal plane changes from small to large in the range of 
[0,2  ], and the corresponding beam sector rotates to form 
a cone with the ship as the vertex, which intersects with the 
seafloor slope to form an ellipse. The different diameters Wβ 
of this ellipse are the detection widths corresponding to 



 

81 

different pinch angles β. When β is 0°, 45°, 90°, 135°, 180°, 
225°, 270°, 315°, the corresponding Wβ is: 


















55.416
05.416

55.415

27090

31522513545

1800

WW
WWWW

WW
   (10) 

The distance dX traveled by the boat in the direction of the 
survey line is decomposed into the horizontal distance dZ 
pointing down the slope and the direction dY parallel to the 
slope, i.e: 

cosXZ dd                (11) 

The orthogonal decomposition of the distance dz in the 
horizontal direction pointing to the slope decline is 
decomposed into the distance dP along the direction of the 
slope normal vector and the distance dC perpendicular to the 
slope normal vector, i.e: 

 coscosXP dd              (12) 

From geometric modeling: 

 
2

                    (13) 

This is obtained from the properties of similar triangles: 

'coscos W

W

xH

H 





           (14) 

The opening angle θ of the multibeam transducer is 120°, 
the slope α is 1.5°, the depth of the sea water D0 at the center 
point of the sea area is 120 m, and the spacing of the survey 
lines d is 0.3 nautical miles = 0.3*1852m = 555.6 m, i. e: 
















md

mD

6.555

120
5.1

120

0




               (15) 

The distance d between survey lines is 0.3 nautical miles 
(555.6 m), and the distance from the survey line to the center 
point is 0 m. The coverage widths corresponding to different 
distances from the survey line and the angle β in the direction 
of the survey line are obtained as shown in Figure 4. 

 
Figure 4. Plot of data corresponding to measuring lines of different 

distances 

From Figure 4, when the angle 







2
0

 ， ∪







 

2
2

3
，  , i.e., the ship's has a downward velocity 

component along the slope, the coverage width increases with 
the distance of the measuring ship from the center point of the 

sea area; When the angle 





 

2

3

2
， , i.e., the boat has a 

velocity component that is upward along the slope, the width 
of coverage decreases as the distance of the measuring boat 
from the center point of the sea area increases. 

2.3. Optimization Strategy for Line-of-sight 
Design under Specified Overlap Rates 

Since the overlap cannot exceed 20%, the spatial 
relationship between survey lines should be parallel. Design 
a set of survey lines with the shortest measuring length that 
can completely cover the entire sea area to be surveyed, i.e., 
it is necessary to determine the direction of the survey lines 
and the spacing of adjacent survey lines. When the vessel 
stays at a certain point, by rotating the direction of the survey, 
a cone with the vessel as the apex can be obtained. The 
intersection of this cone with the seabed slope is an ellipse, 
and the individual diameters of the ellipse are the detection 
widths corresponding to different angles β. 

The shorter the total length of the lines to be measured, the 
larger the detection width corresponding to each line should 
be. The longest diameter in the ellipse is the long axis, so the 
long axis is used as the detection width, i.e., the line is parallel 
to the contour line, as shown in Figure 5. 

 
Figure 5. Schematic of the profile 

 
From the given information, the shallowest part of the sea 

(east side) is two nautical miles from the center, then the depth 
of sea water at the shallowest part is Dmin: 

tan1852*2min  ChD          (16) 
Similarly, the deepest seawater depth Dmax is: 

tan1852*2max  ChD          (17) 
To cover the shallowest part of the sea area, the first 

sounding line sea depth D1 on the east side is the deepest: 


tan

2
tanminmin1 DDD            (18) 

To cover the deepest part of the sea area, the first western 
sounding line sea depth DS is the shallowest: 


tan

2
tanmaxmax DDDS          (19) 

Meanwhile, for the sake of data accuracy and error 
reparability, the overlap rate should satisfy the requirement of 
10%~20%; meanwhile, in order to satisfy the shortest total 
measurement length, the overlap rate η=10% is selected. 

The opening angle θ of the multibeam transducer is 120°, 
the slope α is 1.5°, and the depth of seawater hc at the center 
point of the sea is 110 m, i.e: 











mhC 110
5.1

120




               (20) 

The coordinates of the starting point and the depth of the 
sea water at the starting point are used as initial values, and 



 

82 

then the distribution of the survey lines is derived as shown in 
Figure 6. 

 
Figure 6. Schematic diagram of the distribution of survey lines 

 
The total length of the survey lines is 75,422.292km, and 

the overlap rate is 10.6%, which is in line with the condition 
that the overlap rate is between 10% and 20%, and it is also 
found that the survey lines are more dense in the "shallow 
water area", and the arrangement of the survey lines is more 
sparse in the "deep water area". 

2.4. Optimization Strategy for Line-of-sight 
Design Considering Submarine Surfaces 

In order to have a clearer and more intuitive understanding 
of the seabed topography of the sea area, the given data were 
visualized. 

Since the surface of the seabed terrain should be segmented 
into a patchwork of multiple planes, the dividing line of each 
plane needs to be determined. And Sobel filter is a 
convolutional filter commonly used in image processing to 
detect edges and contours in an image. It calculates the 
gradient value of a pixel by applying two 3*3 convolution 
kernels to each pixel of the image, which can be used to 
calculate the horizontal and vertical gradients of the image, 
which are used to identify the edges in the image, and then to 
determine how to segment the surface into planes. 

 
Figure 7. Depth gradient map of the seafloor 

 
The gradient approximation of the seafloor depth data is 

obtained by two-dimensional convolution of the Sobel filter 

with the seafloor depth data, and the gradient threshold is set 
to 2.588, i.e., the coordinate points with gradient greater than 
2.588 are marked in purple, as shown in Figure 7. 

Analysis of the gradient diagram shows that there can be 
two ways of dividing the seabed surface, method one is the 
square area division and method two is the triangular area 
division centered on the diagonal. Both method one and 
method two use the difference between the mean values of the 
depths of the two ends of the plane to calculate the slope angle 
of each plane. After calculation, the total length of the 
measured line of method I is 441044.819m, Leakrate1=5.2%; 
the total length of the measured line of method II is 
325906.828m, Leakrate1=15.7%. By comparing the total line 
lengths, the total line length of Law II is shorter, the 
percentage of missed sea area to the total sea area to be 
measured is larger, and the total length of the part with overlap 
rate more than 20% is shorter. Method 1 has a longer total line 
length, a smaller percentage of the total sea area to be 
measured, and a longer total length for the portion of the line 
with an overlap of more than 20%. It was found that fitting a 
plane to a surface results in a loss of accuracy, and therefore 
results in an increase in the rate of missed measurements or 
the total length of the survey line. 

3. Model Summary 

3.1. Model Benefits 
1) Based on geometric analysis, this paper establishes an 

ideal geometric model and an equal-depth iterative model, 
which explains the problem in an easy-to-understand way, 
provides a tool for accurate reasoning, and is able to solve the 
core query effectively. 

2) The numerical solutions derived from this model in the 
application process are more in line with the actual situation 
and can reasonably describe the bathymetric process of the 
multibeam sidelobe system. 

3) The structure of this model is clear and simple, and its 
visualization is beautiful and easy to distinguish. 

3.2. Model Shortcomings 
1) This model needs to use approximation processing when 

dealing with the seabed surface, which splits the surface into 
multiple planes, and there is a certain error. 

2) There is a certain error in the ideal geometric model, and 
in real life, because sound waves bend in water, the geometric 
model can only approximate the composition of the triangular 
relationship, and thus the calculated numerical solution will 
have a certain error. 

4. Conclusion 
Based on the multibeam bathymetric system, this paper 

establishes an ideal geometric model and an isobath iterative 
model. The actual physical problem of multibeam bathymetry 
is abstracted into a geometrical model about a conic curve 
through the physical method of ideal model, and the 
coordinates of the next line are iterated according to the 
current depth under the constraints of the overlap rate, thus 
solving for the optimal set of lines. First, a mathematical 
model of the coverage width and the overlap rate between 
adjacent bathymetric strips for multibeam bathymetry is 
established, and the seawater depths, coverage widths, and 
overlap rates with the previous survey line corresponding to 
survey lines at different distances from the centroid are 
calculated according to this model. Then, the angle between 



 

83 

the direction of the survey line and the normal direction of the 
seafloor slope projected on the horizontal plane is introduced, 
and the similar triangle property is utilized to solve the 
coverage widths corresponding to the survey lines at different 
distances and the angle β. Again, the optimal survey line is 
designed, the shortest total survey line is solved by isobath 
iterative modeling, and the designed combination of survey 
lines is verified to ensure complete coverage of the entire sea 
area. Finally, the seafloor terrain is segmented into different 
planes with slope constraints by Sobel edge detection, and 
then survey lines are designed for each plane, and the spacing 
between survey lines is adjusted by the isobath iterative 
algorithm to meet the overlap rate requirement. The structure 
of the proposed mathematical model is clear and simple, 
which is in line with the actual situation. 

References 
[1] Yu Qiyi. Seabed topography survey based on multibeam 

bathymetry. Surveying, Mapping and Spatial Geographic 
Information, 2022, 45(9): 262-264. 

[2] Zhu Huimin. Key technology and accuracy assessment of 
underwater topographic measurement by multibeam 
bathymetric system. Longitude and Latitude World, 2022(2): 
4-6. 

[3] Yang, Haidong. Analysis of the influence of multibeam 
sounder attitude on bathymetry and error correction techniques. 
Applied Acoustics: 1-15 [2023-09-10]. 

[4] Jaewoo J, Youngtak K, Joohan L, et al. Multibeam Bathymetry 
and Distribution of Clay Minerals on Surface Sediments of a 
Small Bay in Terra Nova Bay, Antarctica. Minerals,2021,11(1). 

[5] Dudkov YI, Sivkov VV, Dorokhov VD, et al. Multibeam 
bathymetry data from the Kane Gap and south-eastern part of 
the Canary Basin (Eastern tropical Atlantic). Data in Brief, 
2020, 32.