Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 8, No. 2, 2023 

 

44 

Model‐Driven	Multibeam	Line	Measurement	System	
Design	
Weiliang Pang1, * 

1School of Mathematics and Information Science, Zhongyuan University of Technology, Zhengzhou, China 
*Corresponding author: 2783247105@qq.com 

 

Abstract: Multibeam bathymetric system is a complex combination of multi-sensor system. The multibeam bathymetric 
system is mainly used in the measurement of water depth of waterways, and the scientific layout of the survey line can improve 
the measurement efficiency, and at the same time, truly and perfectly reflect the topography of the waterway. This paper 
establishes a mathematical model of coverage width and overlap rate between adjacent strips for multibeam bathymetry, and 
calculates the parameters such as water depth, coverage width and overlap rate between adjacent strips at the center point of 
different survey lines. Firstly, a single slope multibeam line model was established to calculate the coverage width and the overlap 
rate between adjacent strips of the multibeam bathymetric system, taking into account the influence of the slope of a single slope. 
Then, considering the influence of the angle between the direction of the survey line and the projection of the normal direction 
of the seafloor slope on the horizontal plane, a mathematical model of multibeam bathymetry with variable survey line direction 
is established by downscaling the geometrical relationship in the three-dimensional space to the two-dimensional plane. Again, 
based on the principle of line design, all the lines with overlap rate between 10% and 20% are comprehensively screened, and 
the enumeration method is adopted to find out the optimal lines under each group of conditions. Finally, a three-dimensional 
seabed model is constructed based on the actual sea depth data, and the optimal measurement direction and spacing of the survey 
lines are determined based on the real seabed topographic features, which achieves the goal of minimizing the length of the 
survey lines, and at the same time, ensures that the survey lines can cover the entire sea area to be surveyed, and the iterative 
method is used to search for the final direction of the survey lines and find the optimal solution. 

Keywords: Multibeam Line Measurement System, Full-Coverage Bathymetry, Visualization, Enumeration Method. 
 

1. Introduction 
In recent years, ocean exploration has received increasing 

attention from States. The tools for ocean exploration have 
also been constantly updated, and multibeam sounding 
technology has emerged. Multibeam bathymetric system is a 
system based on the development of single-beam bathymetric 
technology, which uses multiple transducers to 
simultaneously transmit acoustic signals and receive signals 
reflected back from the seabed on the receiving transducer. 
The multibeam bathymetric system overcomes the 
shortcomings of single-beam bathymetry and is able to realize 
full-coverage bathymetry [1-4]. In multibeam bathymetric 
systems, the coverage width W varies with the transducer 
opening angle θ and water depth D. If the survey lines are 
parallel to each other and the seafloor terrain is not clear, the 
coverage width W varies. If the survey lines are parallel to 
each other and the seafloor topography is flat, the overlap rate 
between neighboring strips is defined as𝜂 1 𝑑/𝑊 -d/W, 
where d is the spacing between two neighboring survey lines 
and W is the coverage width of the strip. If𝜂 0,  then it 
indicates a missed measurement. In order to ensure the 
convenience of measurement and data integrity, the overlap 
rate between neighboring strips should be 10% to 20%. 
Meanwhile, due to the large undulation of the real seabed 
topography, if the average bathymetry of the sea area is used 
to design the line spacing, although the average overlap rate 
between strips can meet the requirements, there may be 
leakage of measurements in the shallower places; if the 
shallowest bathymetry of the sea area is used to design the 
line spacing, although the overlap rate of the shallowest 
places can meet the requirements, there may be too much 

overlap in deeper places, which will result in a large amount 
of redundancy of the data, affecting the efficiency of the 
measurements [5-8]. 

For this reason, this paper addresses the problem of 
multibeam line sounding, establishes a mathematical model 
of the coverage width and the overlap rate between adjacent 
strips for multibeam sounding, and calculates the parameters 
of water depth, coverage width and the overlap rate between 
adjacent strips at the center point of different sounding lines 
from the center point. Firstly, a single slope multibeam line 
model was established to calculate the coverage width and the 
overlap rate between adjacent strips of the multibeam 
bathymetric system, taking into account the influence of the 
slope of a single slope. Then, considering the influence of the 
angle between the direction of the survey line and the normal 
direction of the seafloor slope projected on the horizontal 
plane, a mathematical model of multibeam bathymetry with 
variable survey line direction is established by downscaling 
the geometric relationship in the three-dimensional space to 
the two-dimensional plane. Again, based on the principle of 
line design, all the lines with overlap rate between 10% and 
20% are comprehensively screened, and the enumeration 
method is adopted to find out the optimal lines under each 
group of conditions. Finally, a three-dimensional seabed 
model was constructed based on the actual bathymetric data, 
and the optimal measurement direction and spacing of the 
survey lines were determined based on the real seabed 
topographic features to achieve the goal of minimizing the 
length of the survey lines, and at the same time to ensure that 
the survey lines could completely cover the entire sea area to 
be surveyed, so that an iterative method could be used to find 
the final direction of the survey lines and find the optimal 



 

45 

solution. 

2. Model Formulation and Solving 

2.1. Single-slope multibeam bathymetric 
modeling 

The seawater coverage of the survey ship's bathymetry is 
given. The coverage width of the multibeam bathymetry is 
affected by the seafloor slope, and the bathymetry model is 
first established at the distance of the survey line from the 
center point, as shown in Figure 1. 

 

 
Figure 1. Schematic diagram of bathymetry at the center 

point 
 

The survey vessel conducts multibeam bathymetry at point 
A. The beams are reflected as they propagate down the slope 
of the seafloor, and the reflection points are noted as B and C. 
From the trigonometric relationship 

 
∠𝐴𝑂𝐶 90∘ 𝛼               (1) 

 
∠𝐴𝑂𝐵 90∘ 𝛼               (2) 

 
Thus the remaining two corners can be derived: 
 

∠𝐴𝐶𝑂 180∘ ∠𝐴𝑂𝐶             (3) 

 

∠𝐴𝐵𝑂 180∘ ∠𝐴𝑂𝐵              (4) 

 
Knowing the depth of the sea water𝐷 70𝑚 , the lengths 

of OC and BO can be found by the sine theorem, viz: 
 

𝑂𝐶
 ∠

sin               (5) 

 

𝐵𝑂
 ∠

sin               (6) 

 
𝐵𝐶 𝐵𝑂 𝑂𝐶                (7) 

 
W is the multibeam strip coverage width and the solution 

equation is: 
 

𝑊 𝐵𝐶 ⋅ cos 𝛼               (8) 
 
Through the above system of equations, it can be concluded 

that the seawater coverage width W is affected by the 
seawater depth D at the current location of the measuring 
vessel, i.e., if the seawater coverage width measured at a 
certain location is required, the seawater depth at the current 
location needs to be clarified. 

Measurement of seawater depths under different survey 

lines with positional relationships as shown in Figure 2. 
 

 
Figure 2. Measured coverage area solving diagram 

 
From the geometric relation: 
 

𝑀𝑁 tan 𝛼 ⋅ 𝑑                (9) 
 
Therefore𝐷 𝐷 𝑀𝑁 , which can be further found as 

𝑊   and similarly one can find𝐷 𝑖 2,3,4   and𝑊 𝑖
2,3,4  . 

The seawater coverage width is the projection of the beam 
to the seafloor level after propagation to the seafloor slope, as 
shown in Figure 3. 

 

 
Figure 3. Schematic diagram of multibeam bathymetry 

 
Also the relationship between coverage and coverage width 

between neighboring strips is: 
 

𝜂 1                  (10) 

 
As the survey vessel takes depth measurements towards 

positions of increasing slope, there will be moments when the 
survey lines do not overlap. 

The solution of the equations is carried out to vary the 
distance of the survey line from the center point to obtain the 
depth of seawater D at different locations, the coverage width 
W, and the coverage η between the current survey line and the 
previous one. 

According to the analysis of single-slope multibeam 
seawater bathymetry data, it can be found that with the center 
point as the starting point, when the bathymetry is carried out 
to the left, the width of the sea area covered by the beam 
gradually increases, and the overlap rate between the adjacent 
lines shows a trend of gradual increase; when the bathymetry 
is carried out to the right of the center point, due to the 
influence of the seafloor slope, the beams of the right side will 
be reflected earlier than the beams of the left side, which 
results in the coverage of the seawater width becomes smaller 



 

46 

and the overlap rate between the adjacent lines gradually 
decreases. The overlap rate between neighboring lines 
gradually decreases. The specific results are shown in Figure 
4. The overlap rate of the survey line is negative when the 
distance from the center is 600m and 800m, that is, the 
phenomenon of missing side. According to the theoretical 
analysis of the average measuring line distance, it can be 
inferred that the model is basically consistent with the 
theoretical model, so it can be considered that the model is in 
line with the actual situation. 

 

 

 
Figure 4. Schematic diagram of the analysis of the results 

at different bathymetric positions 

2.2. Multibeam bathymetric model with 
variable line direction 

Multibeam bathymetry system is developed on the basis of 
single-beam bathymetry system, which can launch multiple 
beams for bathymetry at a time in the plane perpendicular to 
the trajectory. Now, according to the question two, assuming 
that a vessel conducts a line survey from point P to point Q on 
the sea level, i.e., the direction of PQ is the direction of the 
line survey, the projection of the slope surface at point P is 
intersected by the slope surface of the seabed at point P', and 
the projection of the horizontal plane is intersected by the 
horizontal plane at point P "point, the same way as the 
projection of point Q in the seabed slope surface and the 
projection of the horizontal plane were intersected with the 
seabed slope surface and the horizontal plane at point Q' and 
Q", from the title we can know that the angle β for the 
direction of the line of measurement and the seabed slope 
surface of the normal projection between the horizontal plane, 
so the normal in the horizontal surface of the projection is 
perpendicular to the seabed slope and the horizontal plane of 
the intersection of the line MN. MN. By the working principle 
of multibeam bathymetry, it can be obtained that multiple 
beams can be sent out when the survey ship works in the plane 
perpendicular to the track, thus the vertical plane of the course 
track PQ is made to intersect the horizontal plane at the point 
S, and P "S is connected to it, and the positional relationship 

in the three-dimensional space is shown in Figure 5. 
 

 
Figure 5. Measured position relationship in three-

dimensional space 
 

During the work of the survey vessel, it can be derived that 
β is the projection of the normal vector of the slope on the 
horizontal plane𝑃 𝐹  The angle between P "F and the line 
connecting the projection of the course PQ on the horizontal 
plane, i.e.𝑃 𝐹 the angle between the projection of the normal 
vector of the slope on the horizontal plane and𝑃 𝑄   " the 
angle between. 

As a result, the measured position relationship under three-
dimensional space is analyzed and solved, due to the spatial 
complexity of the three-dimensional relationship, this 
question is solved by first adopting the dimensionality 
reduction processing, so that the three-dimensional spatial 
relationship is projected into a two-dimensional position 
relationship, first of all, we can learn that the point P' is the 
projection of the point P on the seabed slope, then in 
theΔ𝑃 𝑃 𝐹  center. We can see that LP'FP" is equal to the 
slope angle α of the seabed slope, assuming that the distance 
between P'P" is X, i.e., the distance between the slope 
projection of P point and the horizontal projection is X. 

Therefore, inΔP P F in, it follows from the sine theorem: 
 

𝑃 𝐹                 (11) 

 
Up to this point, the ΔP FS  downscaled to a two-

dimensional graph. 
The positional relationship between the angle β between 

the direction of the survey line and the projection of the 
normal direction of the seafloor slope in the horizontal plane 
can be derived from the fact that the angle β is affected by the 
direction of the survey line, and because multibeam 
bathymetry works by transmitting beams under the vertical 
plane of the course. It can be derived from the triangular 
relationship: 

 

P S
 ∘               (12) 

 
From the above equation, we can derive the 𝑃 𝑆  The 

geometric relationship between the angle β and the angle β, at 
the same time𝑃 𝑆 ⊥ 𝑃 𝑄 , The projected line of two points 
P, Q in the horizontal plane, and when the measuring ship is 
sailing, because the multi-beam needs to transmit beams in 
the course vertical plane, so the course vertical plane will 
intersect the seabed slope in a line, and the angle between the 
intersecting slope and the horizontal plane is affected by the 
β-angle, so the slope of the intersecting slope is assumed to 
be y, and at the same time, the intersecting slope is 
downgraded to make it in the two-dimensional plane for the 
Solving. It is obtained by the sine theorem: 



 

47 

𝛾 arctan tan             (13) 

 
Here, we have utilized the dimensionality reduction 

technique to simplify the working position relationship of the 
survey vessel, which was originally in three-dimensional 
space, to a planar position relationship. Therefore, we can 
construct the multibeam bathymetry coverage model of this 
question based on the model conclusion in question one. 
According to Fig. 6-6, we derive the relationship expression 
for the seawater coverage width as: 

 

𝑊 𝐷
 ∘

 
𝐷

 ∘

 
cos 𝛾        (14) 

 
The expression for the overlap rate relationship between 

neighboring bands is: 
 

𝜂 1                  (15) 

 
where θ is the transducer opening angle, d is the spacing 

between neighboring strips, and γ is the slope. 
The above system of equations was solved, and the 

coverage width W was calculated using the multibeam 
bathymetric coverage model by calculating the angle β 
between different survey lines and the projection of the 
seafloor slope normal in the horizontal plane, and by taking 
into account how much distance the survey vessel is from the 
center point, and the results were obtained for different angles 
and positions. 

2.3. Line design model for the sea area to be 
surveyed 

The application of bathymetric modeling to the design of 
survey lines in different directions is explored. By rationally 
planning a specific set of survey lines, the aim is to achieve 
full coverage of a rectangular sea area with a north-south 
length of 2 nautical miles and an east-west width of 4 nautical 
miles. In doing so, we minimized the total length of the lines 
and ensured that the overlap between adjacent strips was kept 
in the range of 10 to 20%. To achieve this overlap rate 
requirement, we traversed and tested all possible overlap rates. 
Based on the model in Problem 2, we can derive the coverage 
width calculation formula as: 

 

𝑊 2 ⋅ 𝐷 ⋅ tan             (16) 

 
The strip spacing can be derived from the formula for the 

overlap rate: 
 

𝑑 𝑊 ⋅ 1 𝜂               (17) 
 
Based on the strip spacing and the length of the sea area to 

be measured, we can derive the number of strips needed to 
measure the sea area to be measured: 

𝑛 𝑐𝑒𝑖𝑙                  (18) 

 
where ceil is the upward rounding function; L is the length 

of the sea area to be measured. 
At the same time, since the coverage width W is affected 

by the seafloor slope, for which we establish an adjustment 
factor correction: 

 

𝐷 𝐷 𝑑 ⋅ tan 𝛼              (19) 
 
At this point, the survey line design model is completed, 

according to the listed system of equations to solve the 
number and location of the survey line, which ensures that the 
survey line can cover the entire sea area to be measured. 

Solving the equations yields the relationship between the 
depth of seawater at different distances, as shown in Figure 6. 

 

 
Figure 6. Map of changes in seafloor depth 

 
At this point, it is necessary to ensure that the overlap of 

the lines is maintained at between 10 and 20 per cent, and 
the lines are traversed at 1 per cent intervals to obtain the 
line spacing and the number of lines required. It is 
necessary to minimize the total length of the lines and to 
ensure that the lines completely cover the sea area to be 
measured. After calculation, the coverage width of the 
target line is 381.05 meters, and the line spacing is 323.89 
meters. In order to cover the entire sea area to be surveyed, 
23 lines need to be laid. 

Based on the spacing of the survey lines and the 
coverage width of the survey lines, in order to express the 
measurement effect of the survey lines more intuitively, we 
visualize all the survey lines, and the obtained results are 
shown in Figure 7. 

 

 
Figure 7. Three-dimensional schematic diagram of line 

measurement 
 

By solving the survey line design model, we can 
conclude that the optimal coverage width of the survey line 
is: 381.05 meters Spacing of the survey line is: 323.89 



 

48 

meters Number of survey lines: 23 Compared with the 
survey lines in other directions, it saves the total length of 
the survey, and at the same time the overlap rate of the 
survey line is: 15%, which is in line with the overlap range 
of the survey line from 10% to 20%, and the working 
length of the survey line is: Spacing of the survey line - 
Number of survey lines, which is greater than the length of 
the sea area to be surveyed, then it is considered that the 
line can completely cover the entire sea area to be surveyed. 
The working length of the line is: line spacing - number of 
lines, which is greater than the length of the sea area to be 
measured, then it is considered that the line can completely 
cover the entire sea area to be measured. 

2.4. Measurement wiring design for multibeam 
wave measurement ships 

To solve the ideal survey line based on the specific 
bathymetric data map, the 3D spatial model of the 
bathymetric data was firstly visualized based on the actual 
bathymetric data as shown in Figure 8. 

 

 
Figure 8. 3D simulation of sea depth 

 
Based on this, we optimize all feasible styli to obtain the 

optimal stylus. Through the enumeration method, we traverse 
all the survey lines and find out the optimal survey line that 
meets the requirements by experimenting with different 
spacing and direction of the survey line , and then calculate 
the total length of the survey line by using the geometric 
relationship. By solving the problem, the optimal line can be 
obtained from the actual bathymetry data, and the optimal line 
is visualized as shown in Figure 9. 

 

 
Figure 9. Simulation of the optimal measurement line 

 

Based on the solution analysis, the relevant parameters of 
the optimal survey line are obtained: 

The angle between the optimal survey line and the 
projection of the slope normal in the horizontal plane is: 0°. 

The optimum line spacing is 0.3 nautical miles. 
The total length of the shortest survey line is: 85,192 meters. 
Define the total length of the survey line, i.e. the sum of the 

lengths of all the survey lines; the spacing of the survey lines 
𝑑 𝐿  This is the distance between two neighboring lines; the 
number of lines, i.e. the number of lines needed to cover the 
sea area to be measured.𝑛  The quantitative relationship can 
be derived: 

 
𝐿 𝑑 ⋅ 𝑛                (20) 

 
We first define the area of the uncovered sea water region 

as𝑊  , the total area of seawater to be measured is W. The 
percentage of the missed sea area is denoted as Q%, so it 
follows from the physical relationship: 

 
𝑄% 𝑊 /𝑊 ⋅ 100%           (21) 

 
First of all, according to the requirements of the topic, we 

need to calculate the width covered by each line, and then 
calculate the overlap area and overlap rate of the lines based 
on the coverage width of these lines, thus defining the overlap 
rate as P%, the width of the overlapped area as N, and the 
coverage area of the lines as M, which can be derived from 
the physical relationship: 

 
𝑃% 𝑁/𝑀 ⋅ 100%           (22) 

 
The total length of the portion of the overlap area in which 

the overlap exceeds 20% is calculated, so based on the portion 
of the above equation where P is greater than 20, i.e: 

Excess of 20% over overlap = (P% - 20%) - width of line 
coverage 𝑃% 20%  

The required portion is the sum of the portion that exceeds 
the 20% overlap rate. 

Calculated based on the above physical equation: 
The total length of the minimum survey line is 85,192 

meters; the percentage of the omitted sea area to the total sea 
area to be surveyed is 1.49 per cent; and the total length of the 
overlapping area with an overlap of more than 20 per cent is 
74,302.24 meters. 

3. Conclusion 
This paper addresses the problem of multibeam bathymetry, 

establishes a mathematical model of coverage width and 
overlap rate between adjacent strips for multibeam 
bathymetry, and calculates the water depth, coverage width 
and overlap rate between adjacent strips at the center of 
different survey lines. Firstly, a single slope multibeam line 
model was established to calculate the coverage width and the 
overlap rate between adjacent strips of the multibeam 
bathymetric system, taking into account the influence of the 
slope of a single slope. Then, considering the influence of the 
angle between the direction of the survey line and the normal 
direction of the seafloor slope projected on the horizontal 
plane, a mathematical model of multibeam bathymetry with 
variable survey line direction is established by downscaling 
the geometrical relationship in the three-dimensional space to 
the two-dimensional plane. Again, based on the principle of 



 

49 

line design, all the lines with overlap rate between 10% and 
20% are comprehensively screened, and the enumeration 
method is adopted to find out the optimal lines under each 
group of conditions. Finally, a three-dimensional seabed 
model is constructed based on the actual sea depth data, and 
the optimal measurement direction and spacing of the survey 
lines are determined based on the real seabed topographic 
features, which achieves the goal of minimizing the length of 
the survey lines, and at the same time, ensures that the survey 
lines can cover the entire sea area to be surveyed, and the 
iterative method is used to search for the final direction of the 
survey lines and find the optimal solution. 

References 
[1] Cheng F, Hu N. Research on the optimization method of 

multibeam survey line laying. Journal of Marine Technology, 
2016, 35(02): 87-91. 

[2] Li F, Jin S, Bian G, Cui Y, Tang Y, Zhang Y. Construction and 
validation of multibeam bathymetry error improvement model. 
Journal of Surveying and Mapping, 2022,51(05):762-771. 

[3] Han X. Planned survey line placement of multibeam sweep 
survey system in coastal engineering construction. Jiangxi 
Surveying and Mapping, 2013(04):19-20. 

[4] Shu S, Cheng L. Analysis on the application of multibeam 
system in Yangtze River waterway survey. China Water 
Transportation, 2017(09):46-47. 

[5] Zhang X, Ye C, Hong D. Research on survey line laying 
method of multibeam system in Yangtze River waterway 
survey. China Water Transportation. Waterway Science and 
Technology, 2017(01):52-55. 

[6] Zhang H. Research on high-precision multibeam bathymetry 
method. Shanghai: Shanghai Ocean University, 2021. 

[7] Wang S, Liu X, Cao J, Zhao H, Wu M. Research and realization 
of multi-strip detection mode of deepwater multibeam 
bathymetric sonar. Ocean Surveying and Mapping, 2022, 
42(02):22-27. 

[8] Chen R, Liu X, Liu Z, Liu X, Dong F. A multibeam bathymetry 
method based on cross-swing stabilization. Acoustic 
Technology, 2013,32(05):368-372.