Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 10, No. 3, 2024 

 

111 

Construction	of	line	Layout	Model	Based	on	Geometric	
Deduction	and	Constrained	Heuristic	Search 
Jinghao Deng, Shunjiao Xue, Yinghan Zhang, Qi Zhang 

North China Electric Power University, Department of Electrical Engineering, Baoding Hebei 071000, China 

 

Abstract: Based on problem B of the National Mathematical Contest in Modeling for College Students in 2023, this paper 
establishes a model through geometric deductive deduction, and establishes and solves an optimal model of survey line layout 
through constrained heuristic algorithm. In problem 1, a mathematical model of coverage width and overlap rate is established 
based on triangle relation and triangle sine theorem by finding the mathematical relationship between target quantity and known 
quantity. In problem 2, the angle between the known direction of the survey line and the projection of the normal direction of 
the submarine slope on the horizontal plane is 𝛽. We can imagine a surface perpendicular to the direction of the measurement 
line, and then find the slope of the imaginary surface according to the geometric relationship, and replace the slope Angle of 
problem 1 with the imaginary slope Angle to obtain the mathematical model of the coverage width. Based on this, the constraints 
of the Angle and overlap rate of the line are added, and the problem 3 is simplified into the optimization problem of multiple 
routes in problem 2.  Finally, by changing the parameter setting of slope Angle in the model, the sensitivity analysis is carried 
out, and it is found that the model is stable and reliable.To sum up, the problems from one to four are gradually deepened and 
progressively advanced, and the development of Marine survey field is facilitated by establishing a model of depth survey line 
layout. 

Keywords: Geometric relation, heuristic algorithm multi-objective programming, overlay width, overlap rate, surface integral. 
 

1. Introduction 
With the urgent need for high-precision seabed topographic 

survey, ocean exploration has developed from discrete, low-
precision and low-efficiency to full coverage, high-precision 
and high-efficiency [1][2]. The traditional single beam 
sounding can not meet the requirement of high efficiency 
seabed exploration. On this basis, the multi-beam sounding 
system has been proposed and developed continuously, and 
has been widely used as a new technology of ocean 
exploration. The layout of line is a very important link in the 
technical design of sea area, but at present, the actual wiring 
design methods in China mostly rely on experience. Based on 
the current situation, the research of multi-beam line 
measurement is particularly important. There are two key 
problems in the layout of multi-beam line. The first is the 
survey line interval. Shallow water depth is used to design the 
survey line interval, and the problem of missing measurement 
may occur in the shallow place[3]. Using deep water depth 
design line interval, there will be more overlapping areas, and 
the data redundancy is large, which affects the efficiency of 
the line. The other is the selection of the direction of the 
measurement line. Different directions of the measurement 
line will affect the difficulty of the measurement line [4]. 

2. Problem Restatement  

2.1. Solution to Problem 1 
Considering the existence of a certain slope on the sea floor, 

the slope 𝛼 is defined when the plane perpendicular to the 
direction of the survey line forms a diagonal line with an 
Angle of 𝛼 from the horizontal plane. To find the expression 
of the coverage width 𝑊  and the overlap rate 𝜂  between 
adjacent strips of multi-beam sounding. According to the 
mathematical expression, the coverage width and overlap rate 
under the given parameters are solved by substitution [5]. 

2.2. Solution to problem 2 
In a rectangular sea area, when the Angle between the 

direction of the survey line and the projection of the normal 
direction of the submarine slope on the horizontal plane is 
arbitrarily determined (recorded as 𝛽 ), the mathematical 
expression of the coverage width is required, and the value of 
the coverage width is obtained with several groups of given 
parameters. 

2.3. Solution to Problem 3 
In a given rectangular sea area, a planning model is 

established, which can cover the entire sea area under the 
constraints required in the question, and the overlap rate 
between adjacent strips is between 10% and 20%. A set of 
optimal measuring lines is designed to make the total 
measuring line length shortest. 

2.4. Solution to Problem 4 
On the basis of some existing reference data, simulate the 

bottom of the sea area, and make appropriate simplified 
processing according to the bottom of the sea area. Under 
constrained conditions, cover the sea area to be measured as 
much as possible, keep the overlap rate of adjacent strips 
below 20%, minimize the total length of the side line as much 
as possible, search for the approximate optimal solution, and 
calculate relevant indicators according to the optimal solution: 
The total length of the measuring line, the proportion of the 
missing measuring area, the total length of the overlap of 
more than 20%. 

3. Problem Analysis 

3.1. Analysis of Problem 1 
The first problem requires a mathematical model of 

coverage width and overlap rate. The key point of this 



 

112 

problem is to find the mathematical relationship between the 
target quantity and the known quantity in plane geometry 
problems. In this regard, according to the triangle relation and 
the sine theorem of triangle, the mathematical expressions for 
covering width 𝑊, overlap rate 𝜂, slope 𝛼, opening Angle of 
multi-beam converter 𝜃 and seawater depth 𝐷 at the center 
of sea area are established respectively [6]. 

3.2. Analysis of problem 2 
Problem 2 is extended to some extent on the basis of 

problem 1. In problem 2, the angle between the direction of 
the survey line and the projection of the normal direction of 
the submarine slope on the horizontal plane is 𝛽 .In fact, 
problem 2 is reduced to the situation of problem one when 
𝛽 90°  or 𝛽 270° . Problem 1 gives a mathematical 
model of coverage width and overlap rate on the surface 
perpendicular to the direction of the measurement line. In 
problem 2, we can imagine a surface perpendicular to the 
direction of the measurement line, and then find the slope of 
the imaginary surface according to the geometric relationship, 
and replace the slope Angle of problem 1 with the imaginary 
slope Angle to obtain the mathematical model of coverage 
width. 

3.3. Analysis of problem 3 
Problem 3 is a problem on optimization, which needs to 

design a set of routes with the shortest measurement length, 
and can completely cover the entire sea area and meet the 
requirement of 10%~20% overlap rate. According to problem 
2, a route can be determined by the Angle 𝛽 projected on the 
horizontal plane between the direction of the survey line and 
the normal direction of the submarine slope (hereinafter 
referred to as the direction of the route) and the interval d of 
the route. Therefore, the problem is simplified to the 
optimization problem of solving the two routes in multiple 
problems. (1) The total length of the survey lines is required 
to be the shortest. When the course direction is fixed, the 
survey line interval 𝑑 is as large as possible, and the number 
of survey lines is as small as possible. Therefore, the overlap 
rate of survey lines at the shallowest part of the sea water is 
at least 10%, and the overlap rate at the deepest part of the sea 
water is not more than 20%; When there is overlap between 
each line, it will definitely meet the complete coverage of the 
sea area between the survey lines, we only need to consider 
the interval from the most marginal survey line to the edge of 
the sea area 𝑑 When the first survey line is determined, the 
second survey line will be determined solely by the first 
survey line according to the maximum interval, and so on 
until all survey lines have scanned the entire sea area [7]. 

3.4. Analysis of Problem 4 
Firstly, the data given in the question is visualized. In 

MATLAB, a three-dimensional map is made. Considering 
that the data is a plurality of data points, in order to make that 
undersea topography more intuitive and smooth, fitting the 
given data, a fitted surface equation is obtained. By looking 
at the contour plot and gradient plot of the surface, it is found 
that that contour line on both sides along the diagonal line are 
nearly parallel and approximately straight, and the gradient 
direction is basically the same. Considering that the sea floor 
on either side of the diagonal along the rectangular sea area is 
approximately planar, the original surface is then decomposed 
into two planes, and the question is reduced to question three. 
According to the title, the mathematical model of multi-

objective programming is established and solved[8]. 
 

 

4. Model Assumption  
1. According to the relevant literature in the field of Marine 

survey, it is generally possible to arrange parallel routes.  
2. The measurement lines laid out are straight lines.  
3. The distance between the surveying ship and the center 

point of the sea area is only considered as the distance 
between the surveying ship and the center point of the sea area 
in the vertical direction of the surveying line, which means, 
the distance between the surveying ship and the center point 
of the sea area along the direction of the surveying line is not 
considered.  

4. Since the propagation speed of sound wave in water is 
much higher than the speed of ship, it is considered that there 
is no time delay effect during each measurement. 

5. Symbol Description 
Table 1. Symbol Description 

symbol Instructions unit 
𝑊 Covering width meter 

𝜃 
Multi-beam transducer opening 

Angle 
Angle 

𝛼 submarine slope Angle Angle 
𝜂 Overlap rate —— 

𝐷 
Depth of sea water at the center 

point of sea area 
meter 

𝑑 
Distance between adjacent survey 

lines 
meter 

𝑙  The i line  
𝛼 Total length of measuring line Angle 
𝐿  Article i Measuring line length meter 
Z Total length of measuring line meter 

𝑥 
Distance between survey line and 

center point of sea area 
meter 

𝐷  Sea depth meter 
Note: Some length units in the question are in nautical 

miles, and all length units in this paper are in the International 
standard unit meter. The specific conversion method is given 
as 1 nautical mile =1852 meters according to the question. 



 

113 

6. Model Establishment and Solution 

6.1. Establishment and solution of problem-1 
model 

6.1.1. Model establishment 
The first problem is to calculate the coverage width and 

overlap rate in a slanted case. Since the question does not 
specify the specific calculation method, we are analogous to 
the horizontal case. 

Define the coverage width as the length of the edge beam 
falling on the seafloor slope at the opening Angle of the 
transducer. 

As shown in Figure 1 below, it is defined as 
 

Covering width W= KB (1) 

 
Define the overlap ratio (denoted 𝜂) as 1 minus the ratio of 

the difference between the transducer coverage width and the 
overlap to the transducer coverage width. 

In Figure 1 below, it is defined as 
 

 

 
Figure 1. Illustration of the first question 

 

6.1.2. Model solution 
Using the sine theorem of triangle and the relation of Angle 

between parallel lines, the mathematical relation is 
established. 

Using the sine theorem and triangle induction formula in 
the triangle AHK, we get: 

 

|𝐾𝐻|
𝑠𝑖𝑛 180° ∠1

|𝐾𝐻|
𝑠𝑖𝑛 ∠1

|𝐷𝑡𝑎𝑛
𝜃
2 |

𝑠𝑖𝑛 ∠𝐴𝐾𝐻
 (3) 

 
Using the sine theorem in the triangle BHL, we get: 
 

|𝐵𝐻|
𝑠𝑖𝑛 ∠1

|𝐷𝑡𝑎𝑛
𝜃
2 |

𝑠𝑖𝑛 ∠𝐻𝐵𝐿
 (4) 

From the parallel line theorem and the sum of the angles of 
a triangle being dark 180° : 

 

∠1 90° 𝜃
2

 (5) 

 

∠𝐴𝐾𝐻 ∠1 ∠𝛼 
(6) 

 
 

∠𝐻𝐵𝐿 180° ∠1 ∠𝛼 
(7) 

 
The quantitative relationships in length are: 
 

𝑊 |𝐴𝐵| |𝐾𝐻| |𝐻𝐵| (8) 
 
Combine formula (3) - (8) and simplify to obtain the 

mathematical expression of covering width  ܹ◌ : 
 

𝑊
2𝐷𝑡𝑎𝑛

𝜃
2 𝑐𝑜𝑠

𝜃
2 𝑠𝑖𝑛 90° ∠𝛼

𝑠𝑖𝑛 90° ∠𝛼
𝜃
2 𝑠𝑖𝑛 90° ∠𝛼

𝜃
2

 (9) 

 
Using the sine theorem in the triangle KEC, we get: 
 

|𝐾𝐶|
𝑠𝑖𝑛 ∠𝐾𝐸𝐶

|𝐸𝐶|
𝑠𝑖𝑛 ∠𝐸𝐾𝐶

|𝑑|
𝑠𝑖𝑛 ∠𝐸𝐾𝐶

 (10) 

 
∠𝐸𝐾𝐶 ∠𝐴𝐾𝐻 (11) 

 
∠𝐾𝐸𝐶 180° ∠1 (12) 

 
Then substitute formula (9) - (12) into formula (2) to obtain 

the expression of overlap rate 𝜂 : 
 

𝜂=1−𝑑𝑠𝑖𝑛 [90°−∠𝛼+𝜃2]2𝐷𝑠𝑖𝑛𝜃2𝑐𝑜𝑠∠𝛼 (13) 
 
Formulas (9) and (13) give the expression when the 

measurement line is 0 from the center point of the sea area. 
When the surveying line moves, the surveying line and the 
center point of the sea area change, and only the seawater 
depth 𝐷 will change (become 𝐷  obscure). Assuming that 
the distance between the surveying line and the center point 
of the sea area is x, then: 

 
𝐷 𝐷 𝑥𝑡𝑎𝑛∠𝛼 (14) 

 
By synthesizing formulas (9), (13) and (14), we can 

achieve the results obtained from the calculation of the given 
parameters in topic 1 through MATLAB programming (see 
Table 2): 

 
Table 2. The result of question 1 

Measure the distance 
between the line and 
the center point /m 

-800 -600 -400 -200 0 200 400 600 800 

Sea depth /m 49.051 54.288 59.526 64.763 70.000 75.237 80.474 85.712 90.949 
Coverage width /m 170.32 188.51 206.69 224.88 243.07 261.25 279.44 297.62 315.81 

Overlap rate with the 
previous line /% 

— -23.0% -11.2% -1.4% 6.8% 13.8% 19.8% 25.0% 29.6% 

 
 

Overlap rate 𝜂 1
| |

 (2) 



 

114 

 

6.2. Establishment and solution of problem 2 
model 

Problem 2 introduces a more complex scenario on the basis 
of problem 1. Since the beam surface is actually a plane 
perpendicular to the survey line, we are more concerned about 
the change of the vertical surface of the survey line in the 
following analysis. In problem 1, the vertical plane of the line 
is parallel to the horizontal component of the slope's normal 
direction, while in problem 2, there is a certain Angle between 
the vertical plane of the line and the horizontal component of 
the slope's normal direction. If the position of the vertical 
plane can be determined, we can imagine a slope formed by 
the vertical plane of the vertical line and the reference plane 
of the seabed, and then solve the slope Angle of the imaginary 
slope surface α ̃ by using geometric relations. Then problem 
2 can be simplified to problem 1. 

Therefore, the center point of the sea area (denoted C) is 
the vertical plane of the survey line and intersects it with the 
submarine reference plane. The intersection line is denoted b. 
Points O and B are at the intersection line b, connect C and O, 
so that CO is perpendicular to the submarine reference plane 
(denoted line segment CO is line h). Point B is at the 
intersection line between the submarine reference plane and 
the bottom slope of the real sea 7 (denoted l). Cross the center 
point C of the sea area to the intersection line of the submarine 
slope and the submarine reference plane as A vertical line. 
The intersection point is denoted as point A, connecting OA 
(denoted as line a), BC. 

The scenario diagram is shown in Figure 2 below 
 

 
Figure 2.  

 
It is easy to know from the figure that there is a 

trigonometric function relationship at this time: 
 

𝑡𝑎𝑛∠𝛼
|𝐶𝑂|
|𝐴𝑂|

，𝑡𝑎𝑛∠𝛼
|𝐶𝑂|
|𝐵𝑂|

 (15) 

 
Since 𝐶𝑂 is the reference plane of the ocean floor,𝐶𝑂 ⊥ 𝑙. 

And since 𝐶𝐴 ⊥ 𝑙，𝐶𝐴 ∩ 𝐶𝑂 𝐶. From the vertical theorem 
of line and plane, it can be concluded that 𝑙 ⊥ 𝐹𝑎𝑐𝑒 𝑂𝐶𝐴. 

Since the line plane is perpendicular,𝑙 is perpendicular to 
any line in the face OCA, i.e. : 

 
∀𝑙𝑖𝑛𝑒𝑥 ⊂ 𝑓𝑎𝑐𝑒 𝑂𝐶𝐴，𝑙 ⊥ 𝑥 (16) 

 
So the line OA is perpendicular to the intersection line 𝑙, 

i.e.,∠𝑂𝐴𝐵 90° 
In the triangle OAB,∠𝑂𝐴𝐵 90°，∠𝐴𝑂𝐵 ∠𝛽 90°, so 

 

𝑐𝑜𝑠∠𝐴𝑂𝐵 𝑐𝑜𝑠 ∠𝛽 90° |𝐴𝑂|
|𝐵𝑂|

 (17) 

 
By combining equations (14) and (16), the mathematical 

relationship of imaginary slope Angle can be obtained: 
 

∠𝛼 𝑎𝑟𝑐𝑡𝑎𝑛 |𝑠𝑖𝑛∠𝛽| 𝑡𝑎𝑛∠𝛼  (18) 
 
After the imaginary slope Angle is obtained, problem 2 is 

simplified to problem 1. ∠𝛼 in formula (9) and formula (14) 
is replaced by ∠𝛼, and 𝐷 in formula (9) is replaced by𝐷 ̵ in 
formula (14). The total formula for the coverage width of 
problem 2 is as follows: 

 

𝑊
2 𝐷 𝑥𝑡𝑎𝑛∠𝛼 𝑡𝑎𝑛

𝜃
2 𝑐𝑜𝑠

𝜃
2 𝑠𝑖𝑛 90° ∠𝛼

𝑠𝑖𝑛 90° ∠𝛼
𝜃
2 𝑠𝑖𝑛 90° ∠𝛼

𝜃
2

 (19) 

 
The formula (19) was programmed in MATLAB, and the 

parameters given in question 2 were substituted, and the 
results were obtained. 

6.3. Establishment and solution of problem 3 
model 

This problem is an optimization problem, which needs to 
design a set of routes with the shortest measurement length. It 
can completely cover the entire sea area, and meet the 
requirement of overlap rate between 10% and 20%.  

It can be seen from problem 2 that a route can be 
determined by the Angle 𝛽 projected on the horizontal plane 
between the direction of the survey line and the normal 
direction of the submarine slope (hereinafter referred to as the 
route direction) and the interval d of the route. The slope is 
1.5° and the opening Angle of the multi-beam transducer is 
120°, which is the same as problem 2. Therefore, the problem 
is simplified to the solution optimization problem of routes in 
problem 2.  

By searching the data, we learned that the survey lines need 
to be arranged in parallel in order to ensure that a set of survey 
lines can scan the entire sea area as fully as possible [9].  

 (1)The total length of the survey lines is required to be the 
shortest. When the course direction is fixed, the survey line 
interval d is as large as possible, and the number of survey 
lines is as small as possible. Therefore, the overlap rate of 
survey lines at the shallowest part of the sea water is at least 
10%, and the overlap rate at the deepest part of the sea water 
is not more than 20%;(2) When there is overlap between each 
line, it will definitely meet the complete coverage of the sea 
area between the survey lines, we only need to consider the 
interval from the most marginal survey line to the edge of the 
sea area 𝑑 . When the first survey line is determined, the 
second survey line will be determined solely by the first 
survey line according to the maximum interval, and so on 
until all survey lines have scanned the entire sea area[10]. 
Step1. Solve for the initial interval 𝑑  
 



 

115 

 
Figure 3.  

 
Figure 3 shows the side view perpendicular to the direction 

of the survey line and passing the edge point C. As shown in 
Figure 3, when the scanning area of the edge survey line 
passes the most edge point C, the entire scanning of the edge 
area can be guaranteed. To maximize the spacing 𝑑  , we 
make the 𝑙   obscure right side scan width 𝑤   obscure 
exactly to the point C on the edge, where the deepest sea water 
is ℎ , there is 

 

 
Step2. Establish a plane rectangular coordinate system for 

the top view of the entire sea area 

 
Figure 4 

 
As shown in FIG. 4, the positive direction of the X-axis is 

set to be westward. According to the meaning of the question, 
in the rectangular sea area 2 nautical miles long from north to 
south and 4 nautical miles wide from east to west, the west is 
deep and the east is shallow. The Angle between the axis of 
the straight line 𝑙  and 𝑥  is the included Angle of the 
horizontal projection of the direction of the survey line and 
the normal direction of the submarine slope, 𝛽, and the Angle 
𝛽 is arbitrary. 

Step3. Solve the upper intersection point 𝐴  and the lower 
intersection point 𝐴  of the survey line 𝑙  and the sea area 
to be measured 

We set intersection 𝐴 𝑥 , 𝑦  , 𝐵 𝑥 , 𝑦   when the 
initial interval grey is determined, then you can write in figure 
4 𝑙  dark in the coordinate system for the linear equation 

 
𝑌 𝑡𝑎𝑛𝛽 ∙ 𝑥 𝑥  (21) 

 
Among them, 
 

 
The equation for line BC is 
 

 
The equation of line AB is 
 

 
If the equations (18) and (20) are combined, a point 

coordinate will be obtained, set as 𝐴 ; if the equations (18) 
and (21) are simultaneously combined, a point coordinate will 
be obtained, set as 𝐴 , which is obtained by comparing the 
horizontal coordinates of 𝐴  and 𝐴  

 

𝑚𝑖𝑛 𝑥 , 𝑥  
(25) 
 

𝐿 𝑥 𝑥 𝑦 𝑦  (26) 
 
Then the upper intersection is the coordinate of the point 

with the smallest horizontal coordinate among the two points, 
and the upper intersection 𝐴   is obtained. Similarly, write 
the equation of a line AO and a line OC to obtain the 
coordinates of the lower intersection, then the length of the 
measurement line is the Euclidean distance between the upper 
and lower intersection points, that is, write the equation of a 
line AO and a line OC to obtain the coordinates of the lower 
intersection points, then the length of the measurement line is 
the Euclidean distance between the upper and lower 
intersection points, that is 

Step4. According to the measurement line 𝑙   obscurity 
and constraints, find out the interval 𝑑   with the next 
measurement line 

From the overlap rate formula (13) obtained in question 1, 
the relationship between interval d and overlap rate 𝜂 is 

 

𝑑
2 ∙ 1 𝜂 ∙ 𝐷 ∙ sin 𝜃/2 ∙ 𝑐𝑜𝑠𝛼

sin 90° 𝛼 𝜃/2
 (27) 

 
Here is the hypothetical slope angle 𝛼 in Problem 2, and 
 

𝑡𝑎𝑛𝛼 𝑡𝑎𝑛𝛼 ∙ 𝑠𝑖𝑛𝛽 (28) 
 
As can be seen from the figure, the depth of the sea area at 

any point is only related to the abscissa of the point 𝑥, and 
the relation is satisfied 

 
𝐷 𝑓 𝑥 ℎ ℎ 𝑡𝑎𝑛𝛼 ∙ 𝑥 (29) 

 
When the route is obscure at the shallowest sea depth 

𝐴 ,𝜂 10% can ensure that the route spacing is as large as 
possible, and 𝐵   should be satisfied when the route is 
obscure at the deepest sea depth 𝜂 20%; 

Step5. Determine that the last route covers the diagonal 
edge point A 

Suppose there are n measuring lines to complete the full 
coverage, repeat Step1-4, and the expression of the 𝑖 
measuring line is. 

 

𝑑 ℎ ∙ tan（𝜃/2） (20) 

𝑥 𝑥 𝑑 /𝑠𝑖𝑛𝛽 (22) 

𝑥 4 ∙ 1852 (23) 

𝑦 2 ∙ 1852 (24) 



 

116 

 
Find the final interval 𝑑  on the basis of the last line𝑙 , if 

the last intersection 𝐴  and the abscissa 𝑥  are satisfied 
 

 
Then 
 

 
On the contrary, 
 

 
Finally, the total length Z of the whole group of 

measurement lines is 
 

 
Again according to question 2, when 0° 𝛽 180° with 

180° 𝛽 360°  when routes, is just the starting point is 
different, so the subject need to consider 0° 𝛽 180° 
situation, above all, kinds of optimization objective function 
for the model 

𝑚𝑖𝑛 𝑍 𝐿  (35) 

 
The constraint condition is 
 

s.t. 

10% 𝜂 20%
0 𝑥 4 1852,     𝑖 1,2,3 …
0 𝑦 2 1852,     𝑖 1,2,3 …

0° 𝛽 180°

 (36) 

 
The final result is that the number of measured lines is 38, 

and the total length is 14,0752m. Draw the group of lines with 
the shortest total distance as follows 

 

 
Figure 5a 

 

 
Figure 5b 

6.4. Establishment and solution of problem 4 
model 

6.4.1. Data Visualization 
We first visualized the data given in the question and made 

a three-dimensional diagram in MATLAB, as shown in Figure 
5. Considering that the data are multiple data points, in order 
to make the submarine terrain more intuitive and smooth, 
cftool toolbox in MATLAB was used to fit the data given, and 
the surface diagram shown in Figure 6 can be obtained. 

 
Figure 6 

And the equation of submarine surface is 
 
𝑧 3.173 17.6 ∙ 𝑥 4.491 ∙ 𝑦 6.4 ∙ 𝑥

9.6 ∙ 𝑥 ∙ 𝑦 3.2 ∙ 𝑦^2 
(37) 

 
And the correlation coefficient 𝑅 0.9648 and the 

goodness of fit is good. 

6.4.2. Model establishment 
In the survey line group with the shortest total distance 

obtained from question 3, the direction of the survey line is 
parallel to the bottom edge of the rectangular sea area, that is, 
it runs along the contour line of the seabed. The feasibility of 
this operation is verified by searching data, and we specify the 
course direction of question 4 to run along the contour line of 
the coastal bottom [11]. 

According to the equation of the submarine surface 
obtained in z, we make the contour map and gradient line of 
the submarine surface as shown in FIG. 7. 

 

 
Figure 7 

 
By analyzing the obtained contour map, we find that the 

𝑌
𝑑

𝑐𝑜𝑠𝛽
𝑌  (30) 

𝑥
𝑑

sin 𝛽
 (31) 

𝑛 𝑖 (32) 

𝑛 𝑖 1 (33) 

𝑍 𝐿  (34) 



 

117 

contour lines on the left and right sides of the diagonal line 
from southwest to northeast in the rectangular sea area are 
basically parallel and approximately straight. At the same 
time, we draw the gradient map and find that the direction of 
the gradients on both sides is basically the same along the 
main diagonal, indicating that the two sides can be regarded 
as planes approximately. 

Inspired by this, the seafloor surface can be divided into 
two planes along the southwest to northeast diagonal, so 
problem 4 is simplified to problem 3. The difference is that 
the seafloor slope of Problem 3 is a rectangular plane, while 
problem 4 is two triangular planes[12]. 

The seafloor plane above the diagonal from southwest to 
northeast is 𝑆  dark, and the seafloor plane below is 𝑆 . At 
the same time, the east direction is the positive direction of 
the x axis, the north direction is the positive direction of the y 
axis, and the upward direction is the positive direction of the 
z axis. The space rectangular coordinate system is established 
with the seabed in the southwest corner as the coordinate 
origin, as shown in Figure 8. 

 

 
Figure 8 

 
As shown in the figure, ∆𝐴𝑂𝐵 belongs to the plane 𝑆 , 

∆𝐴𝐶𝐵  belongs to the plane 𝑆  . Since problem 3 is a 
rectangular slope and problem 4 is a triangle, it is considered 
to put the triangle into the rectangular slope to solve it. Taking 
the plane 𝑆  as the example ∆𝐴𝑂𝐵, extend the base of AB 
to point E and connect to OE, where OE is the base edge of 
the rectangular slope to be obtained. 

Then the problem is simplified to problem 3. We arrange 
survey lines parallel to the base edge OE (that is, in the 
direction of contour lines), and finally increase the constraints 
that the total length of survey lines is as short as possible, the 
coverage area is as large as possible, and the overlap rate is as 
small as possible to establish a new optimization model. 

A rectangular area things for a long, north and south long 
b,，𝐴 𝑥 , 𝑦 , 𝑧 ,𝐵 𝑥 , 𝑦 , 𝑧 , 𝐶 𝑥 , 𝑦 , 𝑧 , 𝐸 𝑥 , 𝑦 , 𝑧 . 
Step1. Find the slope Angle α 

The direction of the vector 𝑂�⃗� is specified as the direction 
of the measurement line, then the Angle 𝛼  of slope is the 
dihedral Angle between the plane 𝑆  and the plane 𝑥𝑂𝑦, and 
a plane 𝑆  is set.If the normal vector of the plane 𝑥𝑂𝑦 is 𝒏, 
then there is 

 

𝒎 𝑶𝑨 𝑶�⃗� 
(38) 
 

𝒏 𝟎, 𝟎, 𝟏  (39) 
 
Dihedral Angle α is 

 

𝑐𝑜𝑠 𝛼
𝒎 𝒏

|𝒎| |𝒏|
 (40) 

 
Step2. Find out the bottom width of the rectangular area 
According to the meaning of the question, OE is the base 

width of the corresponding rectangular area, and the 
schematic diagram of the rectangular sea area with S  as the 
slope is shown in FIG. 9. 

 

 
Figure 9 

 
Then there is, 
 

 
Step3. Find the bottom length of the rectangular area 
As shown in FIG. 10, the vertical line intersecting the base 

of xOy through point A meets the point P x , y , 0 , then the 
length of the base OM is the distance from point P to the 
straight line OE. In the coordinate system plane xOy , the 
equation of the straight line OE is 

 
And 

 
Then the distance from point P to line OE is 
 

𝑑
𝑘𝑥 𝑦

√𝑘 1
 

(44) 

 

 
Step4. Regression problem 3 
The expression formula of the overlapping area is as 

follows 
 

𝑆 𝜂 ∙ 𝑊 ∙ 𝐿  (45) 

 
In summary, the newly obtained mathematical model S  

is 
 

⎩
⎪
⎨

⎪
⎧min 𝑆 𝜂 ∙ 𝑊 ∙ 𝐿

min 𝑍 𝐿

 (46) 

 
The constraint condition is 
 

|𝑂𝐸|= 𝑥 𝑦  (41) 

𝑦 𝑘 ∙ 𝑥 (42) 

𝑘
𝑦
𝑥

 (43) 



 

118 

s.t. 

𝜂 20%
0 𝑥 𝑂𝐸,     𝑖 1,2,3 …
0 𝑦 𝑑,     𝑖 1,2,3 …

𝛽 90°

 (47) 

 
The procedure for S  is the same as above and will not be 

described here. 
The result of question 4 is finally obtained, as shown in 

Table 3. 
 

Table 3. The result of question 4 

 
 
As can be seen from the table, the final total number of 

measured lines is 50, the optimal length is 469255.51m, the 
area of overlapping area is 0. The area of unmeasured area is 
17895500.19𝑚  , accounting for 0.42% of the total surface 
area, and the result is good [13]. 

7. Model Analysis and Test 

7.1. Model sensitivity analysis 
Sensitivity analysis is one of the methods to evaluate the 

planning model, which is as follows[14] : 
(1) Set a bias for the required parameters of the model (take 

3%, 6%, 9% as an example). 
(2) When the slope Angle of the parameter  is changed, 

the remaining parameters are kept unchanged to observe 
whether the difference between the final results is significant. 
The results are as follows 

 
Table 4. The results 

 
 
The relationship between the change of slope Angle and the 

number of lines and the total length of lines is shown in Figure 
10. 

 

 
Figure 10a 

 

 
Figure 10b 

 
The results show that when the parameters change, the 

model results will not change greatly, and the model stability 
is reliable. 

7.2. Model error analysis 
The model assumes that the delay caused by the beam 

propagation time does not affect the results, which was 
reviewed: The propagation speed of sound waves in seawater. 
The main factors of the propagation speed of sound waves are 
temperature, salinity and pressure (depth). The change of 
temperature has the greatest influence on sound velocity. Due 
to the uneven distribution of temperature and salinity in the 
seawater medium, the distribution of sound velocity will be 
uneven, thus forming a sound velocity gradient in the ocean. 
The speed of sound wave propagating in the seawater is three 
and a half times faster than that in the air (about 1531 m/s)[15]. 

We estimate the speed of sound at 1 nautical mile per 
second, and the maximum time for the beam to sweep the 
rectangular sea area is not more than 5 seconds, so it is easy 
to know that this assumption will not introduce significant 
errors. 

8. Evaluation, Improvement and 
Extension of The Model 

8.1. Advantages of the model 
1. The model is based on the integrity of hydrographic 

survey and has practical application prospects.  
2. Some assumptions are made in the model to simplify the 

problem and reduce the amount of calculation.  
3. The model is based on geometric ideas and simple to 

prove.  
4. The model qualitatively analyzes the effects of acoustic 

wave propagation to simplify the problem. 

8.2. Disadvantages of the model 
1. Problem 4 model retains the integrity of measurement as 

much as possible, and the measurement line obtained is too 
long.  

2. In the modeling of problem 4, we chose the fitting graph 
instead of the actual surface, which may result in the loss of 
some data points. 

8.3. Model improvement 
The line spacing in the model can be assumed to be equal 

spacing, which can simplify the problem and be easy to 
implement, but will sacrifice the accuracy of the model. At 
the same time, for problem 4, we can further refine the 
segmentation of the slope bottom and divide it into more 
planes with different gradients for solving, so as to improve 



 

119 

the accuracy of the model. 

8.4. Model promotion 
1. In this paper, the problem of line layout for multi-beam 

measurement of classical seabed depth is studied, which can 
be applied to more complex seabed sounding problems with 
improvement.  

2. By changing some parameters, a new method can be 
provided for the mapping of various complex maps.  

3. It can be applied to the design of 3D imaging technology 
to scan the physical path. 

References 
[1] Cheng Fang, Hu Niancheng. Research on optimization method 

of multi-beam survey line layout [J]. Journal of Marine 
Technology,2016,35(02):87-91. 

[2] Wang Ning, Huang Jing, Rao Tiantian. Line optimization 
model in the case of multi-beam sounding system [J]. Journal  

of electroacoustic technology, 2023,47 (6) : 61-64. The DOI: 
10.16311 / j.a udioe. 2023.06.018.  

[3] Shallow multi-beam sounding sonar technology innovation and 
application [J]. Ocean development and management, 2012, 29 
(08) : 37, 38, DOI: 10.20016 / j.carol carroll nki hykfygl. 
2012.08.022.  

[4] Hai-sen li, shan, Sunday. Based on spatial smoothing of multi-
beam sounding sonar coherent distributed source azimuth 
estimation [J]. Journal of vibration and shock, 2014 (4) : 138-
142. The DOI: 10.13465 / j.carol carroll nki JVS. 2014.04.024. 

[5] Zheng Peng, zhang hua, zi-ying pan, etc. Multi-beam sounding 
sonar imaging simulation and range analysis [J]. Journal of 
digital ocean and underwater offensive and defensive, 2023, 6 
(6) : 664-669. The DOI: 10.19838 / j.i SSN. 2096-
5753.2023.06.003.  

[6] Yang Jian. Sonar technology application in Marine resources 
exploration and development [J]. Journal of electroacoustic 
technology, 2023,47 (12) : 27-29. DOI: 10.16311 / j.a udioe. 
2023.12.006.  

[7] Wu Xiaolong. Data Management Platform of Multi-beam 
sounding sonar system [D]. Harbin Engineering 
University,2018. 

[8] SUN Dejun, DENG Yuhang, ZHU Suqian. Multi-beam 
sounding model based on grid processing and ergodic search 
algorithm [J]. Journal of Nantong Vocational 
University,2023,37(04):70-76. 

[9] Kong Zaojie, Operations Research [M], China Machine Press, 
2007.06. 

[10] Code for Hydrographic Survey [M], (GB12327-1988), 1988. 

[11] Zhang Xu, Ye Carefully, Hong Demei. Research on line layout 
method of multi-beam system in Hydrographic survey of 
Yangtze River [J]. China Water Transport. Channel of science 
and technology, 2017 (01) : 52-55, DOI: 10.19412 / j.carol 
carroll nki. 42-1395 / u 2017.01.011.  

[12] Peng Dongdong. Based on multi-beam sounding sonar 
underwater terrain aided navigation and positioning method 
research [D]. Harbin engineering university, 2022. The DOI: 
10.27060 /, dc nki. Ghbcu. 2022.000048.  

[13] The multi-beam sounding system measurement technology 
requirements [M] (JT/T790-2010), 

[14] Yang Lijuan, Zhang Xining, Wang Baojian et al., 2010. 
Development and application of Multi-scale linear 
displacement measurement experimental system -- A case 
study of "Mechanical Engineering Test Technology" course [J]. 
Ministry of Industry and Information Technology education, 
2024 (02) : 71-75 + 90.  

[15] baidu encyclopedia [OL], water velocity and water velocity / 
6118123 https://baike.baidu.com/item/? Fr = ge - ala, 
2023/9/10.