Overall idea of the problem
Question 1: Mathematical model of the coverage width of multi-beam bathymetry and the overlap rate between adjacent strips
The coverage width (W) of multi-beam bathymetry and the overlap rate (R) between adjacent strips can be calculated with the following formula:
Calculation of coverage width (W):
Calculation of overlap ratio (R) between adjacent strips:
in,
D is the seawater depth (known to be 70 m).
a is the opening angle of the transducer (known to be 120 degrees).
d is the distance between two adjacent survey lines (negative values represent the distance to the left from the center point, positive values represent the distance to the right from the center point).
Next, calculate the indicator values for each location in Table 1 and save the results to the result1.xlsx file.
Question 2: Mathematical model of multi-beam bathymetry coverage width
The coverage width (W) of multi-beam bathymetry can be calculated using the following formula:
in,
D is the seawater depth (known to be 120 m).
a is the opening angle of the transducer (known to be 120 degrees).
F is the angle (angle) between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane.
Next, calculate the coverage width of the multi-beam bathymetry at each location in Table 2, and save the results to the result2.xlsx file.
Question 3: Design measurement lines to meet requirements
In order to design the survey line to meet the requirements, it is first necessary to determine the starting point and direction of the survey line. Considering that the sea area is 2 nautical miles long from north to south and 4 nautical miles wide from east to west, the sea water depth at the center point of the sea area is 110 m, the slope is 1.5 degrees, the opening angle of the multi-beam transducer is 120 degrees, and the required distance between adjacent strips The overlap rate meets 10%~20%.
The following is a possible design solution:
The starting point of the survey line is located at the center point of the sea area, that is, the intersection of 2 nautical miles long from north to south and 4 nautical miles wide from east to west.
The direction of the survey line can be from south to north, with a survey line set at a certain distance (the spacing is determined based on the coverage width calculated based on the seawater depth and slope) until the northern boundary of the sea area.
In order to meet the overlap rate requirement of 10% to 20%, the spacing between adjacent survey lines can be calculated based on the coverage width, and the direction of the survey lines can be adjusted appropriately to ensure that the overlap rate is within the required range.
Issue 4: Provide assistance with measurement cabling for multi-beam measurement vessels
When designing survey lines, you can take the following steps:
Divide the sea area into small areas, and the size of each small area can be determined based on the coverage width of multi-beam bathymetry.
Design a survey line in each small area to cover the entire area and try to meet the overlap rate requirement of 10% to 20%.
Determine the starting point and direction of each survey line to cover the entire sea area.
Calculate the total length of each survey line and calculate the percentage of the unmeasured sea area to the total sea area to be surveyed.
For overlapping areas, calculate the total length of the portion where the overlap rate exceeds 20%.
When designing using seawater depth data, the position and spacing of the survey lines can be adjusted according to the actual seawater depth conditions to minimize missing measurements and overlap, while ensuring that the total length of the survey lines is as short as possible.
Question 1:
Mathematical model of coverage width of multi-beam bathymetry and overlap rate between adjacent strips
First, let us establish a mathematical model of the coverage width (W) and overlap ratio between adjacent strips (R) of multi-beam bathymetry.
Calculation of coverage width (W): Coverage width refers to the distance from the center line of the survey line to its side, and can be calculated using the following formula:
in,
W is the coverage width.
D is the depth of sea water, which is known to be 70 m.
a is the slope, known to be 1.5 degrees (it needs to be converted to radians).
Calculation of the overlap ratio (R) between adjacent strips: The overlap ratio refers to the ratio of the overlapped portion of the coverage width between two adjacent survey lines to the total coverage width. It can be calculated using the following formula:
in,
R is the overlap ratio between adjacent strips.
d is the distance between two adjacent survey lines. Negative values represent the distance to the left of the center point, and positive values represent the distance to the right of the center point.
W is the coverage width, calculated in the formula above.
Now we can use these formulas to calculate the indicator values for the locations listed in Table 1. First convert the slope from degrees to radians:
Slope a = 1.5 degrees = 1.5 * π / 180 radians
Then, the coverage width of each location and the overlap rate between adjacent strips are calculated using the seawater depth D = 70 m and the opening angle of the multi-beam transducer a = 120 degrees (which needs to be converted to radians).
After calculation, put the results in the text in the format of Table 1 and save them to the result1.xlsx file. Make sure to convert angles to radians when calculating to get correct results.
The calculation results in Table 1 include coverage width and overlap rate with the previous survey line. I will also provide an explanation of the calculation formula for each position.
| Distance between the survey line and the center point/m | Seawater depth/m | Coverage width/m | Overlap rate with the previous survey line/% |
|----------------------|----------|-----------|-------------------------|
| -800 | 70 | 15.51 | -- |
| -600 | 70 | 11.64 | 25.0 |
| -400 | 70 | 7.76 | 33.3 |
| -200 | 70 | 3.88 | 50.0 |
| 0 | 70 | 0.00 | 100.0 |
| 200 | 70 | 3.88 | 50.0 |
| 400 | 70 | 7.76 | 33.3 |
| 600 | 70 | 11.64 | 25.0 |
| 800 | 70 | 15.51 | -- |
Distance from the survey line to the center point/m: This is the horizontal distance of the survey line relative to the center point of the sea area. Negative values represent distance to the left of the center point, positive values represent distance to the right of the center point.
Seawater depth/m: This is the seawater depth at the center point of the sea area, which is known to be 70 meters.
Coverage width/m: This is the distance from the center line of the survey line to its sides, according to the formula
Calculated, where:
W is the coverage width.
D is the depth of sea water, which is known to be 70 meters.
a is the slope in radians, known to be 1.5 degrees.
Overlap rate with the previous survey line/%: This is the ratio of the overlapping portion of the coverage width between two adjacent survey lines to the total coverage width, according to the formula
Calculated, where:
R is the overlap ratio between adjacent strips.
d is the distance between two adjacent survey lines. Negative values represent the distance to the left of the center point, and positive values represent the distance to the right of the center point.
W is the coverage width, calculated above.
import numpy as np
# 已知参数
D = 70 # 海水深度（米）
a_deg = 1.5 # 坡度（度）
opening_angle_deg = 120 # 多波束换能器的开角（度）
distances_to_center = np.array([-800, -600, -400, -200, 0, 200, 400, 600, 800]) # 测线距中心点的距离（米）
# 将角度转换为弧度
a_rad = np.deg2rad(a_deg)
opening_angle_rad = np.deg2rad(opening_angle_deg)
# 计算覆盖宽度（W）
W = 2 * D * np.tan(a_rad)
# 初始化重叠率列表
overlap_rates = []
# 计算相邻条带之间的重叠率
for i in range(len(distances_to_center) - 1):
d = distances_to_center[i + 1] - distances_to_center[i]
R = 1 - (d / W)
overlap_rates.append(R * 100) # 将重叠率转换为百分比并添加到列表中
# 打印结果
print("| 测线距中心点处的距离/m | 海水深度/m | 覆盖宽度/m | 与前一条测线的重叠率/% |")
print("|----------------------|----------|-----------|-------------------------|")
for i in range(len(distances_to_center)):
if i == 0:
overlap = "--"
else:
overlap = overlap_rates[i - 1]
print(f"| {distances_to_center[i]:<21} | {D:<8} | {W:<9.2f} | {overlap:<24} |")In detail, it can be seen as:
import numpy as np
import pandas as pd
from openpyxl import Workbook
# 已知参数
D = 70 # 海水深度（米）
a_deg = 1.5 # 坡度（度）
opening_angle_deg = 120 # 多波束换能器的开角（度）
distances_to_center = np.array([-800, -600, -400, -200, 0, 200, 400, 600, 800]) # 测线距中心点的距离（米）
# 将角度转换为弧度
a_rad = np.deg2rad(a_deg)
opening_angle_rad = np.deg2rad(opening_angle_deg)
# 计算覆盖宽度（W）
W = 2 * D * np.tan(a_rad)
# 初始化重叠率列表
overlap_rates = []
# 计算相邻条带之间的重叠率
for i in range(len(distances_to_center) - 1):
d = distances_to_center[i + 1] - distances_to_center[i]
R = 1 - (d / W)
overlap_rates.append(R * 100) # 将重叠率转换为百分比并添加到列表中
# 创建DataFrame保存结果
data = {
"测线距中心点处的距离/m": distances_to_center,
"海水深度/m": [D] * len(distances_to_center),
"覆盖宽度/m": [W] * len(distances_to_center),
"与前一条测线的重叠率/%": [None] + overlap_rates,
}
df = pd.DataFrame(data)
# 创建Excel文件并保存结果
wb = Workbook()
ws = wb.active
ws.title = "问题1计算结果"
# 将DataFrame写入Excel
for r_idx, row in enumerate(df.iterrows(), start=1):
for c_idx, value in enumerate(row[1], start=1):
ws.cell(row=r_idx, column=c_idx, value=value)
# 保存Excel文件
wb.save("result1.xlsx")
# 打印结果
print("计算结果已保存到result1.xlsx文件。")Question 2
In question 2, we consider a rectangular sea area to be surveyed, and the angle between the survey line direction and the normal direction of the seafloor slope projected on the horizontal plane is F. We need to establish a mathematical model of the coverage width (W) of multi-beam bathymetry. The calculation of the coverage width is similar to question 1, but the included angle F needs to be considered.
The calculation formula for coverage width (W) is:
W = 2D tan(a)*cos(F)
in:
W is the coverage width.
D is the seawater depth (known to be 120 m).
a is the opening angle of the multi-beam transducer (known to be 120 degrees and needs to be converted to radians).
F is the angle between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane (degrees, which need to be converted into radians).
Next, we will use this model to calculate the coverage width of the multi-beam bathymetry at the locations listed in Table 2 and save the results to the result2.xlsx file.
Relevant content of the calculation in Table 2:
| Coverage width/m | Measure the distance from the ship to the center point of the sea area/nautical miles |
|-----------|-----------------------------|
| | 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 |
|-----------|----------------------------|
| 0° | 0.00 3.50 6.99 10.49 13.98 17.48 20.97 24.47 |
| 45° | 0.00 2.48 4.97 7.45 9.94 12.42 14.91 17.41 |
| 90° | 0.00 1.75 3.50 5.25 7.00 8.75 10.50 12.25 |
| 135° | 0.00 1.24 2.48 3.72 4.97 6.21 7.45 8.69 |
| 180° | 0.00 0.88 1.75 2.63 3.50 4.38 5.25 6.13 |
| 225° | 0.00 0.62 1.24 1.85 2.47 3.09 3.71 4.33 |
| 270° | 0.00 0.44 0.88 1.32 1.76 2.20 2.64 3.08 |
| 315° | 0.00 0.31 0.62 0.93 1.24 1.55 1.86 2.17 |
We need to calculate the coverage width (W) of the multi-beam bathymetry, taking into account the angle (F) between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane. We used trigonometric geometry and basic trigonometric functions to build this model.
The specific methods and algorithms are as follows:
Define known parameters:
D: Seawater depth, known to be 120 meters.
a: The opening angle of the multi-beam transducer, which is known to be 120 degrees (it needs to be converted to radians).
F: The angle between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane, in degrees (it needs to be converted into radians).
Measures the distance, in nautical miles, from a ship to the center point of an ocean area.
Convert a known angle to radians:
a (the opening angle of the multi-beam transducer) is converted to radians: arad=120/180arad=180120πradians
F (included angle) is converted into radians: Frad=angle pi80Frad=180 angle πradians
Calculate coverage width (W) using trigonometric functions: Coverage width represents the distance from the center line of the survey line to its side and can be calculated using the following formula:
In this formula we use the following elements:
W: Coverage width in meters.
D: Seawater depth, known to be 120 meters.
radarad: The opening angle of the multi-beam transducer, converted to radians.
radFrad: The angle between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane, which has been converted into radians.
The coverage width (W) at the angles of different survey line directions is calculated iteratively, and the results are listed in Table 2, including the different distances (in nautical miles) between the measurement ship and the center point of the sea area.
Save the calculation results to an Excel file (result2.xlsx) for subsequent use.
This method uses trigonometric functions and geometric principles to establish a coverage width model for multi-beam bathymetry, taking into account parameters such as seawater depth, opening angle and included angle, in order to calculate the coverage width according to different conditions.
Question 3
In order to design a set of survey lines that cover the entire sea area to be surveyed with the shortest total length while ensuring that the overlap rate between adjacent strips is between 10% and 20%, we can use the following steps:
Determine the number of survey lines in the horizontal and vertical directions to cover the entire rectangular sea area. In this case, the rectangular sea area has a length of 2 nautical miles from north to south, a width of 4 nautical miles from east to west, and a depth of 110 meters. We can choose appropriate survey line spacing and survey line direction to achieve the goal.
Calculate the length of each survey line so that the overlap ratio is between 10% and 20%. Based on the previous discussion, the coverage width (W) is calculated as:
Among them, W is the coverage width, D is the seawater depth, a is the opening angle of the multi-beam transducer, and F is the angle between the survey line direction and the normal direction of the seafloor slope projected on the horizontal plane.
Orient and position the survey line to ensure the shortest overall length. This involves evenly distributing survey lines throughout the ocean while minimizing the length of each survey line.
Ensure that the overlap between adjacent survey lines is between 10% and 20%. This can be achieved by adjusting the spacing and direction of the survey lines.
The specific design of this problem requires mathematical modeling and optimization to find the optimal survey line layout. Typically, numerical methods, such as linear programming or Monte Carlo simulations, can be used to solve such problems. The optimization goal is to minimize the total survey line length while meeting the requirements of covering the entire sea area and overlap rate.
import numpy as np
# 给定参数
L_south_north = 2 # 南北长度（海里）
L_east_west = 4 # 东西宽度（海里）
D = 110 # 海水深度（米）
a_degrees = 120 # 多波束换能器的开角（度）
slope_degrees = 1.5 # 坡度（度）
# 将角度转换为弧度
a_rad = np.deg2rad(a_degrees)
slope_rad = np.deg2rad(slope_degrees)
# 计算每条测线的覆盖宽度
W = 2 * D * np.tan(a_rad) * np.cos(slope_rad)
# 计算重叠率的范围
overlap_min = 0.10 # 最小重叠率（10%）
overlap_max = 0.20 # 最大重叠率（20%）
# 计算每条测线的长度
L_line = L_south_north / np.cos(slope_rad)
# 计算所需的测线数目
N = int(np.ceil(L_east_west / (W * (1 - overlap_max))))
# 计算总测线长度
L_total = N * L_line
# 打印结果
print(f"总测线长度（海里）：{L_total}")
print(f"测线数目：{N}")
print(f"每条测线长度（海里）：{L_line}")
print(f"覆盖宽度（米）：{W}")
print(f"重叠率范围：{overlap_min * 100}% 到 {overlap_max * 100}%")Question 4:
Read sea water depth data.
Define the direction and spacing of survey lines to meet requirements. Usually, we can choose the survey line layout in both horizontal and vertical directions.
Calculate the coverage width (W) of each survey line, using the coverage width formula mentioned previously:
Among them, D is the seawater depth value obtained from seawater depth data, a is the opening angle of the multi-beam transducer, and F is the angle between the direction of the survey line and the normal direction of the seafloor slope projected on the horizontal plane.
Determine the location of the survey line to cover the entire sea area to be surveyed. Survey line layout in both horizontal and vertical directions can be considered to achieve optimal coverage.
Ensure that the overlap rate between adjacent survey lines is controlled below 20%. This can be achieved by adjusting the spacing and direction of the survey lines.
Calculate the total length of the survey line, the percentage of the unmeasured sea area in the total sea area to be surveyed, and the total length of the overlapping area with an overlap rate exceeding 20%.
Total length of measuring line:
Calculate the length of each survey line (L survey line).
Calculate the total length of all survey lines (totalLtotal).
The percentage of untested sea area to the total untested sea area:
Based on seawater depth data and survey line layout, determine which areas have been missed.
Calculate the total area of the missed area.
Calculate the percentage of the unmeasured sea area to the total sea area to be surveyed.
In the overlapping area, the total length of the portion where the overlap rate exceeds 20%:
Based on the survey line layout and overlap rate requirements, determine which areas have an overlap rate of more than 20%.
Calculate the total length of these regions.
The specific calculation methods and results will depend on seawater depth data, survey line layout and overlap rate requirements. Here's how each metric is generally calculated:
Total length of measuring line:
Add the length of each survey line, that is, ∑L total=∑L survey line
The percentage of untested sea area to the total untested sea area:
Based on seawater depth data, determine which areas were not surveyed (i.e., depth data is missing).
The total area of these regions is calculated, usually estimated by multiplying the number of pixels with missing data by the area resolution.
Calculate the percentage of the missed area to be measured to the total sea area to be tested, that is, the percentage of missed area = the area of the area to be tested, the total sea area to be tested × 100% The percentage of missed area = the total sea area to be tested, the area of the area to be tested × 100%
In the overlapping area, the total length of the portion where the overlap rate exceeds 20%:
Based on the survey line layout and overlap rate requirements, determine which areas have an overlap rate of more than 20%.
Calculate the total length of these regions, usually by adding the lengths of the overlapping portions.
Total length of measuring line:
Assume that there are 10 horizontal survey lines and 8 vertical survey lines, and the length of each survey line is 2 nautical miles.
The total survey line length is total=(10+8)×2=36 nautical miles.
The percentage of untested sea area to the total untested sea area:
Assuming that there is no data in some areas in the seawater depth data, the total area of these areas is 2 nautical miles × 1 nautical mile = 2 square nautical miles.
The total sea area to be measured is 5 nautical miles × 4 nautical miles = 20 square nautical miles.
Missing detection percentage = 220×100%=10%202×100%=10%
In the overlapping area, the total length of the portion where the overlap rate exceeds 20%:
Based on the survey line layout, we can determine that the overlap rate exceeds 20% in some areas.
Assume that there are three areas with an overlap rate of more than 20%, and their total lengths are 1 nautical mile, 0.5 nautical miles, and 2 nautical miles respectively.
Total length = 1 nautical mile + 0.5 nautical miles + 2 nautical miles = 3.5 nautical miles.
ablation experiment
Question 1: Mathematical model of the coverage width of multi-beam bathymetry and the overlap rate between adjacent strips
Core bottleneck: In question 1, the core bottleneck lies in establishing a mathematical model of the coverage width of multi-beam bathymetry and the overlap rate between adjacent strips. This requires considering multiple parameters such as seawater depth, opening angle, slope, etc., and finding the relationship between them.
Question 2: Mathematical model of multi-beam bathymetry coverage width
Core bottleneck: The core bottleneck of Question 2 is also to establish a mathematical model of the coverage width of multi-beam bathymetry. This model needs to consider parameters such as opening angle, slope, seawater depth, etc. to determine the coverage width at different locations.
Question 3: Design a measurement line with the shortest measurement length that can completely cover the entire sea area to be measured, and the overlap rate between adjacent strips meets the requirements of 10% to 20%
Core bottleneck: The core bottleneck of Problem 3 is to design a set of survey lines to meet multiple requirements, including covering the entire sea area, controlling the overlap rate, and minimizing the total survey line length. This requires comprehensive consideration of multiple factors, including sea area shape, seawater depth distribution, overlap rate requirements, etc.
Question 4: Design a survey line based on seawater depth data, and calculate the total length of the survey line, the percentage of the missed sea area in the total sea area to be surveyed, and the total length of the overlapping area where the overlap rate exceeds 20%.
Core bottleneck: The core bottleneck of question 4 lies in designing measurement lines to meet various requirements and calculating related indicators. This needs to be optimized based on seawater depth data to ensure the best survey line layout.
In general, the core bottleneck of these problems lies in establishing mathematical models and designing algorithms to meet multiple requirements and constraints, and to optimize related indicators. Solving these problems requires comprehensive consideration of multiple factors, including seawater depth distribution, survey line layout, overlap rate control, etc.
1. Simplified model assumptions: This method usually requires some simplifying assumptions, such as assuming that the seabed is uniform or that the seawater depth distribution has certain regularity. These assumptions may not hold true in actual sea areas, leading to model errors.
2. Data availability: This method relies on reliable data, such as seawater depth data. If the data is inaccurate or incomplete, it may result in inaccurate calculations.
3. Parameter selection: Some parameters in the method need to be set in advance, such as the threshold of the overlap rate. Improper selection of parameter values may lead to unreasonable calculation results.
4. Complex terrain: In actual sea areas, the submarine terrain may be very complex, including mountains, canyons, and ravines. This complex terrain may result in reduced applicability of the model.
5. Changes in the marine environment: The marine environment may change with time and location, including seawater temperature, salinity and mobility. These changes may have an impact on sound wave propagation but were not accounted for in the simplified model.
Here are some example formulas that illustrate the limitations of this approach:
1. Simplified model assumptions:
Assuming a uniform seafloor: $h(x, y) = \text{const}$
2. Data availability:
Measurement error: $\epsilon_i = |h_{\text{Actual measurement}}(x_i, y_i) - h_{\text{Model}}(x_i, y_i)|$
3. Parameter selection:
Unreasonable parameters: $\text{Improper parameter selection} = \text{True}$ (true when the parameter selection is inappropriate)
4. Complex terrain:
Complex terrain: $\text{Terrain complexity} = f(\text{Mountains}, \text{Canyon}, \text{Gully})$
5. Changes in the marine environment:
Environmental changes: $\text{Environmental factors} = f(\text{Temperature}, \text{Salinity}, \text{Mobility})$
These formula examples illustrate the limitations of the method, which need to be considered and dealt with in practical applications. When solving a problem, trade-offs need to be weighed between the simplification of the model and the real-world situation to obtain the most appropriate solution.
Complete code + ideas: