Mathematical modeling of multi-beam line survey problems
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1.Restatement of the problem:
Single-beam bathymetry is a method of measuring water depth using technology that uses sound waves to propagate through water. It calculates water depth by measuring the time it takes for a sound wave to be sent from a ship until it returns. However, since it is measured continuously at a single point, the data is very dense along the track, but there is no data between survey lines.
The multi-beam sounding system is developed on the basis of single-beam sounding. It can send multiple acoustic beams at the same time to cover a wider area. This means that in a flat area on the seabed, it can measure a water depth area with a certain width with the measuring ship’s route as the axis. To ensure data accuracy, there needs to be a certain overlap between adjacent measurement areas.
However, the complex changes in seafloor topography make it a challenge to set the distance between survey lines. If the average water depth is used to set the spacing of survey lines, it may lead to missed measurements in places with shallower water depths. If the shallowest water depth is used to set the spacing, it may lead to excessive overlap in places with deeper water depths, wasting data and time. Therefore, a mathematical model needs to be established to determine the optimal survey line spacing to achieve efficient and accurate measurements under different water depth conditions.
2. Problems to be solved:
Question 1: In single-beam bathymetry, the depth of a body of water is measured by emitting a sound wave and measuring its travel time. The data points are densely distributed along the survey ship trajectory, but there are no data points between trajectories. The problem requires establishing a mathematical model of the coverage width of multi-beam bathymetry and the overlap rate between adjacent strips.
Question 2: Consider a rectangular sea area to be surveyed, where the slope of the seafloor affects the coverage width of multi-beam bathymetry. The problem requires establishing a mathematical model of multi-beam bathymetry coverage width.
Question 3: In a rectangular sea area, a set of survey lines needs to be designed to cover the entire sea area with the shortest length while meeting the overlap rate requirements between adjacent strips.
Question 4: A set of single-beam bathymetry data is provided, and the goal is to design a set of measurement wiring for multi-beam bathymetry. The total length of the survey line is required to be as short as possible while covering the entire sea area to be surveyed and controlling the overlap rate between adjacent strips.
3. Modeling ideas, code, and results
3.1 Question 1
Considering the key parameters of multi-beam bathymetry, namely opening angle, slope and seawater depth, we used the previously established mathematical model to calculate the indicator values for the locations shown in Table 1. These metric values include coverage width W and overlap ratio between adjacent strips. For convenience of reference and recording, we embed the results into the text in tabular form and save them in the result1.xlsx file at the same time. These calculations are crucial to determine optimal multibeam bathymetry parameters and achieve high-quality survey data.
When building the mathematical model for Problem 1, we need to consider several key factors. First, the relationship between the coverage width W, the transducer opening angle θ and the water depth D is a key factor. Second, given the seafloor slope α, we need to determine how the slope affects the coverage width W and the overlap rate n between adjacent strips.
From a geometric point of view, the coverage width W can be expressed as:
Here, W is the coverage width, D is the water depth, and θ is the opening angle of the transducer. This formula shows that W is proportional to D and θ. The deeper the water depth, the larger the opening angle, and the wider the coverage width.
Then, we consider the effect of slope α on the overlap rate n. The overlap rate n can be expressed as:
Among them, d is the distance between two adjacent survey lines. This formula shows that when d is less than W, the overlap rate n is negative, indicating that there is a missed detection.
To meet convenience and data integrity requirements, we need to ensure that n remains between 10% and 20%. Therefore, we can adjust the parameters D, θ, and α to achieve the desired range of overlap rates under different seafloor topographic conditions.
Considering that sound waves are emitted from the water surface and returned after being reflected on the seabed, the coverage width W can be expressed as:
Among them, W is the coverage width, D is the water depth, and 0 is the opening angle of the transducer. This formula describes the path that sound waves travel, thereby determining the coverage width W. The deeper the water depth or the larger the opening angle 0, the wider the coverage width W, because the sound wave takes longer to reflect back from the water bottom, and the wider the seabed area is covered. This formula is very important in multibeam bathymetry because it helps determine the geographical area that can be covered by each measurement
When taking into account the slope α of the seafloor, the actual coverage width is affected by the slope. Slope will cause actual water depth to vary, thus affecting coverage width. Therefore, when considering the slope, the water depth D should be adjusted to D’, where:
This formula corrects the original water depth D to the actual water depth D’, taking into account the influence of seafloor slope. The greater the slope α, the greater the corrected water depth D, and the coverage width W will also be affected. Therefore, in multi-beam bathymetry, the impact of seafloor slope on coverage width must be considered in order to calculate the coverage area more accurately.
Therefore, the coverage width model considering slope can be expressed as:
Among them, W’ is the actual coverage width after considering the slope, D’ is the actual water depth after correction, and θ is the opening angle of the multi-beam transducer. This model takes into account the effect of seafloor slope on coverage width and can be used to calculate the coverage width of a multi-beam bathymetry system under different conditions.
According to the topic description, the overlap rate n can be expressed as:
Among them, n represents the overlap rate between adjacent strips, d is the spacing between two adjacent survey lines, and W’ is the actual coverage width after considering the slope.
First, we need to calculate the seawater depth D at different distances. This can be calculated using the following formula:
Among them, D is the seawater depth from the center point of the measuring ship, D_c is the seawater depth at the center point of the sea area (70 m in the question), y is the horizontal distance from the center point of the measuring ship, α is the slope (in the question is 1.5°).
This formula takes into account the effect of slope on seawater depth, so it can be used to calculate the depth D at different distances.
Next, we can use the seawater depth D at these different distances to calculate the coverage width W. The coverage width W can be calculated using the following formula:
Among them, W is the coverage width, D is the seawater depth from the center point of the measuring ship, and θ is the opening angle of the multi-beam transducer (120° in the question).
This formula takes into account the influence of seawater depth and opening angle on the coverage width, so it can be used to calculate the coverage width W at different distances.
3.2 Question 2
The coverage width W of multi-beam bathymetry can be expressed by the following mathematical model:
W represents the coverage width.
D’ is the water depth corrected according to the slope α.
θ is the opening angle of the multi-beam transducer, which is 120° according to the description of the question.
Part of the code
% Given parameters theta = 120; % opening angle in degrees alpha = 1.5; % slope in degrees center_depth = 120; % depth at the center in meters distances = [0, 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1] * 1852; % distances from the center in meters angles = [0, 45, 90, 135, 180, 225, 270, 315]; % angles in degrees % Function to calculate depth based on the center depth and distance from the center calculate_depth = @(center_depth, alpha, distance) center_depth - tan(deg2rad(alpha)) * distance; % Function to calculate coverage width based on depth and opening angle calculate_width = @(depth, theta) 2 * depth * tan(deg2rad(theta / 2)); % Calculate depths and coverage widths for all combinations of distances and angles depths = calculate_depth(center_depth, alpha, distances); widths = (calculate_width(depths, theta)') * cos(deg2rad(angles)); % Replace negative values with 0 widths(widths < 0) = 0; widths
3.3 Question 3
3.4 Question 4
4. More details and code (private messages welcome)
Detailed code information: https://github.com/HuaandQi/Mathematical-modeling.git