Efficient Weather Routing Method in Coastal and Island-Rich Waters Guided by Ship Trajectory Big Data

Abstract

Weather routing is a critical guarantee for the safe and economical navigation of ships. Existing methods for weather routing still face challenges in selecting the appropriate planning granularity. A granularity that is overly coarse may result in routes passing through coastal and island-rich waters, such as coastal zones and reefs, thus compromising navigational safety. Conversely, a granularity that is excessively fine leads to an exponential increase in computational complexity, rendering the problem intractable. To address this issue, this paper proposes an efficient method for weather routing in coastal and island-rich waters, guided by ship trajectory big data: First, an adaptive quadtree is used to partition the navigable space into an adaptive grid, based on which a route network is constructed using ship trajectory big data. Next, a ship motion model is introduced to build both static and dynamic marine environmental fields, which are used to dynamically update the time weights of the route network. Finally, using the updated route network as a guide, the method aims to minimize voyage time and employs an improved time-varying A* algorithm for weather routing. Experimental results show that the proposed method effectively adapts to coastal and island-rich waters, outperforming the baseline SIMROUTE in safety, optimization, and efficiency. Unlike SIMROUTE, which crosses restricted areas, it avoids such risks entirely. It achieves average reductions of 6.8% in route length and 4.3% in navigation time and is 5.8 times faster than SIMROUTE for fine-grained planning. This balances voyage time, safety, and efficiency, offering a practical weather routing solution.

1. Introduction

Weather routing is a key foundation for ensuring safe and efficient completion of maritime voyages. The marine environment, particularly factors such as water depth, wind, and wave conditions, has a significant impact on both the efficiency and safety of ship navigation. Weather routing refers to the process of optimizing a ship’s route based on navigation tasks (such as time minimization or energy efficiency maximization), while considering the characteristics of the marine area, meteorological forecast data, ship performance, technical conditions, and navigational requirements. The goal is to plan a route that is both safe and time-efficient, as well as energy-saving and cost-effective, from origin to destination [1]. Weather routing plays an essential role in responding to maritime typhoons, rough seas, navigational safety, and reducing casualties, making it a crucial element in ensuring the safe and economical operation of ships. As a result, research on weather routing has attracted widespread attention from scholars.

Weather routing methods could be broadly classified into three categories:

The first is the improved isochronous line method. The isochronous line method was originally a manual approach, centered on constructing an “isochronous line” that represents all possible locations a ship can reach within a predefined time frame. Hagiwara [2] improved the manual isochronous line method by considering environmental factors affecting the voyage and proposed optimizing the ship’s heading and speed to minimize time costs. Klompstra et al. [3] introduced the concept of an “isochronous surface,” using navigation restrictions to define reachable areas. Building on this, Szlapczynska et al. [4,5] reviewed various isochronous line improvement methods and pointed out that in specific geographical environments, such as narrow straits, the method may produce “isochronous loops,” leading to inaccurate routing results and potentially guiding the ship onto dangerous routes. To address this, Roh [6] optimized the isochronous line algorithm to reduce the occurrence of these loops. Lin et al. [7] proposed a three-dimensional isochronous line method, introducing a floating grid system and recursive forward algorithms to compute optimal routes under desired arrival time conditions. Although the improved isochronous line method has successfully transitioned from manual to automated planning and has gradually been applied in weather routing, the inherent “isochronous loop” problem remains unresolved, particularly in specific geographic areas (e.g., narrow straits) where the method is difficult to apply [8].

The second is the artificial intelligence method (AI). In recent years, AI has become a key tool for speed prediction and enhancing navigation efficiency. Wang et al. [9] developed a real-time energy efficiency optimization model based on wavelet neural networks, taking into account environmental factors such as wind speed, water depth, and wake coefficient, to evaluate ship navigation states under different sea conditions. Mao et al. [10] combined ocean environmental data (e.g., wave height, wave period, wind speed) with container ship engine RPM data, applying statistical models such as autoregression, least squares estimation, and maximum likelihood estimation for speed prediction. Gkerekos et al. [11] used artificial neural networks to predict fuel consumption and employed Dijkstra’s algorithm to plan optimal routes. However, AI techniques are currently mainly used for predicting voyage parameters like fuel consumption and engine speed. In recent years, scholars have used machine learning and big data mining to address route planning challenges. Yan et al. [12] identified ship waypoints and formed maritime routes via AIS data, statistical density analysis and DBSCAN, but excluded meteorological factors. In contrast, Vitali et al. [13] integrated voyage and meteorological data to estimate container ship speed loss, filling this gap. For in-depth AIS mining and route detection, Yim et al. [14] built a framework using ship trajectory data and clustering to generate quantitative routes from AIS. Liu et al. [15] further proposed a machine learning framework to extract the global maritime network from AIS, including key processes and an A* algorithm for route planning. Regarding route prediction and design, Lee et al. [16] developed an LSTM-based model using port-to-port data to resolve traditional AIS issues. Zhang et al. [17] proposed an SRU-based method: compressing AIS, extracting turning points via Laplacian eigenmaps and Gaussian kernels, clustering with fuzzy adaptive DBSCAN, and learning ship relationships via SRU to get routes.

The third is the graph search method. Padhy et al. [18] used Dijkstra’s algorithm for weather routing in the North Indian Ocean. Park et al. [19] combined the A* algorithm with speed scheduling to reduce travel time by 2%. Kurosawa et al. [20] proposed a novel A-star-based marine meteorological route planning system, modifying the cost function for oceanic/atmospheric conditions, adding three optimal path search options, and incorporating an unsafe condition avoidance algorithm. Bentin et al. [21] integrated the A* algorithm with the Flettner operator to optimize fuel consumption. Mannarini et al. [22] proposed a numerical calculation model called VISIR for optimal ship routing based on sea condition forecasts, accounting for factors such as wave resistance and stability loss, and improved Dijkstra’s algorithm to enhance navigation efficiency and safety. Through improvements to the A* algorithm, Manel et al. [23] introduced the SIMROUTE method, which effectively improves planning accuracy and navigational efficiency, providing a systematic and efficient engineering solution for weather routing, and has been adopted by the European Union’s Copernicus Programme. Although the SIMROUTE method has been successfully implemented, it still faces challenges in coastal and island-rich waters. This method requires pre-establishing a uniform grid within the navigable area to construct the route network. The key steps involve first determining the topological relationship between the grid and navigational obstacles, and removing overlapping areas, followed by connecting the grids to build the route network. However, the shortcomings of this method are as follows: (1) In long-distance voyages, selecting the appropriate grid scale near complex coastal areas poses challenges, as it directly determines the granularity of the route planning. If the grid scale is too large, the route may intersect with obstacles, compromising navigational safety. Conversely, if the scale is too small, the computational complexity grows exponentially, making the problem unsolvable [23]. To ensure feasible computational results, a slightly larger grid size is typically chosen to optimize computational speed. (2) The constructed regular network connections are overly idealized and may not align with actual ship navigation trajectories. Since the method does not adequately account for real-world complexities, such as ship navigation habits and common route choices, the planned route often fails to reflect the actual sailing path, making it difficult to implement in practice.

To address the aforementioned challenges, this paper proposes an efficient method for weather routing in coastal and island-rich waters, guided by ship Route Network Construction Algorithm. The main contributions and innovations are as follows:

(1)

To address the issue of selecting the appropriate grid scale, which determines the granularity of route planning, this paper introduces an adaptive quadtree-based method for partitioning navigable space into adaptive grids. This approach can dynamically adjust the granularity of the planning based on geographical features. Fine granularity is used in coastal and island-rich waters, while coarser granularity is applied in open seas, thereby improving both the safety and efficiency of the routing.

(2)

To address the issue of excessive computation time in long-distance weather routing, this paper proposes a route network construction method based on historical ship trajectory big data. Building upon the adaptive grid partitioning, the method uses trajectory clustering to construct an empirical route network. It retains frequently used historical routes and eliminates unused ones, further improving the computational efficiency of the route planning.

(3)

The proposed method dynamically integrates marine environmental factors into the route planning process, enabling the generation of navigable, safe, and time-efficient weather routes in coastal and island-rich waters, thereby ensuring both navigation efficiency and safety.

The rest of the paper is organized as follows: As shown in Figure 1, Section 2 presents the adaptive route network construction method based on ship trajectory big data. Section 3 combines marine environmental data with the ship motion model to dynamically update the time weights in the route network. Section 4 discusses weather routing using an improved time-varying A* algorithm, considering dynamic factors. Section 5 provides a case study, validating the effectiveness of the proposed method through simulation tests under two different scenarios. Section 6 concludes the paper, summarizing the main findings and discussing potential directions for future research.

2. Adaptive Route Network Construction Based on Ship Trajectory Big Data

2.1. Adaptive Grid Partitioning of Navigable Space Based on Adaptive Quadtree

Based on the principles of adaptive quadtree partitioning for navigable space, the accuracy of the route network is contingent upon the granularity of the grid division. Excessively high partitioning precision can lead to inefficient planning and computational difficulties, while a low precision may result in a coarse network that compromises safety in coastal and island-rich waters. To address this issue, this paper combines dynamic quadtree segmentation with efficient encoding strategies, dynamically optimizing the size and depth of grid divisions near complex coastal areas based on the specific shapes and locations of navigational hazards. This approach aims to enhance spatial partitioning accuracy adaptively, maximizing the retention of navigable space without sacrificing navigational safety, thereby improving computational efficiency while mitigating risks associated with inappropriate scale selection.

Upon constructing the quadtree, each leaf node corresponds to an independent grid. Subsequently, the navigable space is distinguished through the spatial topological relationship between the leaf nodes and the navigational hazards. Specifically, a recursive algorithm is employed to traverse each node of the quadtree, starting from the root and progressively delving deeper. During this process, a topological check is performed on each node to determine its position relative to the navigational hazards. The classification of node types follows these rules: if a node is entirely within navigable space and has no intersection with the hazard area, it is marked as fully navigable; if a node is entirely within the hazard area or partially overlaps with it, it is marked as hazardous; if a node intersects the boundary of the hazard area and has not reached the maximum partitioning depth, further subdivision is required to ascertain the navigable space more precisely. The aforementioned “further subdivision” operation is governed by the adaptive refinement criterion, which entails the point-by-point insertion of nodes along the boundaries of navigational hazards. This criterion draws on the boundary-adaptive subdivision logic proposed by Zhou et al. [24] in their research on empirical route network mining within complex maritime geographical areas. Specifically, Zhou et al. optimized regional boundaries via iterative point insertion to improve adaptability to complex geographical conditions—a methodological approach that provided a foundational framework for defining the subdivision termination conditions employed in this study (i.e., subdivision ceases when the maximum partitioning depth is attained or when sub-nodes no longer intersect with hazard boundaries).

Next, the boundary areas are refined. For nodes requiring further subdivision, the adaptive grid partitioning strategy continues until the predetermined maximum depth is reached or the node type is clearly defined. Ultimately, all grids marked as navigable will be retained, collectively forming the navigable space grid.

Figure 2a illustrates the navigable space grid generated using adaptive grid partitioning in the maritime area near the Shandong Peninsula, with a longitude range of 117.5–124° E and a latitude range of 35–39° N, resulting in a total of 665 navigable space grids. Figure 2b provides an enlarged view of a local area, where the yellow region represents the hazardous area, and the blue grids denote navigable space. As depicted in Figure 2, the algorithm generates a denser grid near complex coastal and reef areas, while appearing sparser in relatively open waters.

2.2. Route Network Construction Based on Ship Trajectory Big Data

Ship trajectory big data refers to the massive amounts of vessel position and navigation information collected through various technical means (such as satellite positioning, radar, etc.), typically covering dynamic data such as the real-time position, heading, and speed of vessels. The Automatic Identification System (AIS) provides high-frequency, high-precision vessel position data and is widely used in maritime navigation management and monitoring. AIS data is extensive and real-time, and can effectively support the trajectory matching and clustering analysis in complex maritime route planning. Therefore, AIS is selected as the primary source of ship trajectory big data in this study. In the practical application of AIS data, affected by factors such as the data collection environment and transmission links, the data often has messy issues including inconsistent formats, excessive redundant information, and the presence of outliers. If directly used, these problems are likely to lead to deviations in data mining. Therefore, it is essential to conduct preprocessing on the AIS data to achieve data simplification and quality optimization. Based on this, this study refers to and adopts the mature data preprocessing method proposed in Reference [25], and systematically completes the work of cleaning, integrating and simplifying the AIS data, so as to ensure that the data meets the requirements of subsequent applications.

In Section 2.1, the adaptive grid partitioning of navigable space has already been obtained. The next step is to construct the route network based on this partitioning. The methods proposed by Mannarini et al. [22] and Manel et al. [23] both use regular grids with 16-directional connections to generate route networks. However, this approach suffers from issues such as excessive connections, incompatibility with adaptive grids, and overly idealized network connectivity. To address these problems, this paper proposes a method based on historical ship trajectory big data, which performs trajectory matching and clustering after the grid partitioning. This method extracts grid connection relationships that meet actual navigation needs, thus optimizing network connectivity, reducing unnecessary edges, and improving the network’s rationality and adaptability.

To balance computational efficiency and data precision, this paper employs a trajectory mining strategy consisting of the following steps: “trajectory point encoding conversion—node matching and mapping—generating encoded pairs for clustering”. The algorithm pseudocode is shown in Algorithm 1 (Pseudocode Table of Route Network Construction Algorithm):

Algorithm 1: Route Network Construction Based on Ship Trajectory Big Data

Input:  1. Preprocessed AIS trajectory dataset T = {T1, T2,…, Tm}, where Tᵢ is the i-th trajectory containing k trajectory segments {Ti1, Ti2,…, Tik}; each trajectory segment Tij includes start point Sij(xs, ys), end point Eij(xe, ye), and ship attribute Aij (type, speed, heading)  2. Adaptive quadtree model AQT (including the set of adaptive codes C_adap for navigable grids)  3. Full-partition quadtree parameters: area range [x_min, x_max] × [y_min, y_max], number of partition levels n

Output:  - Route network G = (V, E), where V is the navigable grid node (represented by adaptive code) and E is the effective connection between nodes (including connection weight determined by clustering frequency)

// Step 1: Trajectory Point Encoding Conversion (convert latitude and longitude of trajectory segment endpoints to full-partition quadtree codes) Function TrajectoryPointEncoding (T, x_min, x_max, y_min, y_max, n):  EncodedPoints = [] // Store full-partition codes of all trajectory segment endpoints  For each trajectory Ti in T:   For each trajectory segment Tij in Ti:    // Process start point Sij(xs, ys)    rows = floor((ys − y_min)/(y_max − y_min) × 2ⁿ)    cols = floor((xs − x_min)/(x_max − x_min) × 2ⁿ)    quadkeys = CalculateQuadkey(rows, cols, n) // Calculate full-partition code according to Equation (1)    // Process end point Eij(xe, ye)    rowe = floor((ye − y_min)/(y_max − y_min) × 2ⁿ)    cole = floor((xe − x_min)/(x_max − x_min) × 2ⁿ)    quadkeye = CalculateQuadkey(rowe, cole, n)    // Store full-partition codes and attributes of the endpoints of this trajectory segment    EncodedPoints.append((quadkeys, quadkeye, Aij))  Return EncodedPoints // Auxiliary Function: Calculate full-partition quadtree code according to Equation (1) Function CalculateQuadkey(row, col, n):  quadkey = 0  For k from 1 to n (level = n):   bit_row = Extract the (n − k + 1)-th binary digit of integer row // bit(i, x) operation   bit_col = Extract the (n − k + 1)-th binary digit of integer col   quad_digit = 2 * bit_row + bit_col // Calculate the quadtree digit of the current level   quadkey = quadkey + quad_digit × 10(k−1) // Accumulate to generate decimal code  Return quadkey // Step 2: Node Matching and Mapping (map full-partition codes to adaptive quadtree codes) Function NodeMatchingMapping(EncodedPoints, AQT):  AdaptiveEncodedPairs = [] // Store adaptive code pairs (OD pairs) of trajectory segment endpoints  For each (quadkeys, quadkeye, A) in EncodedPoints:   // Map full-partition code quadkeys of the start point to adaptive code   c_adaps = Match(AQT, quadkeys)   // Map full-partition code quadkeye of the end point to adaptive code   c_adape = Match(AQT, quadkeye)   // Store adaptive OD pairs and ship attributes   AdaptiveEncodedPairs.append((c_adaps, c_adape, A))  Return AdaptiveEncodedPairs // Auxiliary Function: Match full-partition code qk in adaptive quadtree AQT and return adaptive code Function Match(AQT, qk):  If qk is in the leaf node codes of AQT:   Return qk // Directly match and return the code  Else:   qk_parent = Obtain the parent node code of qk (move up one level in the full-partition quadtree)   Return Match(AQT, qk_parent) // Recursively match upwards until an existing node is found // Step 3: Generate Encoded Pairs and Clustering (generate effective connections based on spatial proximity and attribute fusion) Function EncodedPairClustering(AdaptiveEncodedPairs):  ConnectionCount = {} // Count the clustering frequency of OD pairs (key: OD pair, value: frequency)  // Step 3.1: Spatial Proximity Calculation (classify identical OD pairs into one category)  For each (cs, ce, A) in AdaptiveEncodedPairs:   od_pair = (cs, ce)   If od_pair not in ConnectionCount:    ConnectionCount[od_pair] = {     “count”: 1,     “attributes”: [A] // Store all ship attributes corresponding to this OD pair    }   Else:    ConnectionCount[od_pair][“count”] += 1    ConnectionCount[od_pair][“attributes”].append(A)    // Step 3.2: Attribute Information Fusion (filter OD pairs with excessive attribute differences to enhance semantic consistency)  ValidConnections = []  For od_pair, data in ConnectionCount.items():   attrs = data[“attributes”]   // Calculate attribute similarity (taking speed and heading as examples; ship types must be consistent)   type_consistent = Whether all ship types in attrs are consistent   speed_similar = Whether the standard deviation of speeds in attrs is less than the set threshold σ_speed   course_similar = Whether the standard deviation of headings in attrs is less than the set threshold σ_course      If type_consistent and speed_similar and course_similar:    // Meet attribute consistency, retain this connection, and the weight is the clustering frequency    ValidConnections.append((od_pair[0], od_pair[1], data[“count”]))    // Step 3.3: Construct Route Network  V = All adaptive code nodes involved in ValidConnections (deduplicated)  E = All ValidConnections (including weights)  G = (V, E)  Return G // Main Process: Call the above functions to complete route network construction Main:  1. Call TrajectoryPointEncoding to generate the set of full-partition codes for endpoints  2. Call NodeMatchingMapping to generate the set of adaptive OD pairs  3. Call EncodedPairClustering to generate route network G  4. Output route network G

(1)

Trajectory Point Encoding Conversion:

The start and end coordinates (latitude and longitude) of each segment of the trajectory are converted into fully partitioned quadtree encodings. The specific formula is as follows [26]:

$$\left\{\begin{array}{c}row=floor({\displaystyle \frac{y-y_{\min }}{y_{\max }-y_{\min }}}\times 2^n)\\ col=floor({\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}\times 2^n)\\ quadkey={\displaystyle \sum _{k=1}^{level}(2\cdot bit(}level-k+1,row)+bit(level-k+1,col))\cdot {10}^{k-1}\end{array}\right.$$

(1)

where (x, y) represents the coordinates of the target point, the boundaries of the region are defined by two vertices: (x_min, y_min) and (x_max, y_max), and n denotes the number of partitions. (row, col) indicates the corresponding row and column numbers, while bit(i, x) is used to extract the i-th binary digit of the integer x. “Level” refers to the depth of the quadtree node corresponding to the trajectory point, and “quadkey” represents the unique encoding of the trajectory point within the fully partitioned quadtree.

(2)

Node Matching and Mapping

After converting the trajectory endpoints’ coordinates into fully partitioned quadtree encodings, the next step is to further convert these into adaptive quadtree encodings. The procedure is as follows: First, the fully partitioned quadtree leaf node encoding of the trajectory endpoints is obtained. Then, in the adaptive quadtree, a matching leaf node encoding is searched. If a matching encoding is found, the mapping is successful; otherwise, a recursive search is performed upwards to the parent nodes until a matching node encoding is found.

(3)

Generating Encoded Pairs and Clustering

The adaptive quadtree encodings of the start and end points of each trajectory segment are combined to generate Origin-Destination (OD) encoded pairs. The clustering process consists of two steps: Spatial Proximity Calculation: The encoded pairs of the endpoints of two segments are compared, and if the encoded pairs are identical, it indicates that the two segments are spatially adjacent and suitable for merging. Attribute Information Fusion: During the clustering process, vessel attributes such as vessel type, speed, and direction are incorporated. By aggregating trajectories with similar attributes, the semantic similarity of the clusters is enhanced, improving the accuracy and practicality of the clustering results. Through the adaptive quadtree partitioning algorithm and segment-based clustering method, both the main shipping routes in the maritime area and the refined secondary routes in complex geographical regions can be captured. A portion of the generated results is illustrated in Figure 3.

3. Dynamic Update of Route Network Weights Integrating Marine Environmental Factors

3.1. Marine Environmental Fields Affecting Ship Navigation

Ships are constrained by both static and dynamic marine environmental fields during navigation. The static environmental field primarily includes various navigational hazards, such as shallow water areas (i.e., regions where the water depth is less than the safe navigation depth for vessels), shipwrecks, and reefs. The dynamic environmental field mainly involves factors like wind, currents, and waves, with waves having a particularly significant impact on ship navigation safety. The information from these environmental fields serves as key input parameters for ship motion models, directly influencing the economic and safety aspects of navigation.

3.1.1. Static Environmental Fields

Obstacles within the static environment can be further categorized into three major types: shallow water hazards, artificial obstacles, and point obstacles. These obstacles pose varying degrees of threat to the safety of ship navigation.

Due to insufficient depth, these areas may lead to vessel grounding or collisions.

(1)

Artificial Hazards: These are specially designated areas, such as no-navigation zones, shipwreck areas, and aquaculture zones. These regions are typically established based on specific navigation rules or safety considerations.

(2)

Point Obstacles: These include individual hazards, such as submerged rocks, unexploded ordnance, and shipwrecks. These point-like obstacles are dispersed independently and pose significant threats to navigation safety. Through nautical chart data, it is possible to obtain and interpret information about these obstacles [27], thereby effectively extracting key elements of navigational hazards. To ensure the reliability of static obstacle information and the safety of subsequent route planning, this paper adopts a buffer zone design for the three types of obstacles (shallow water hazards, artificial obstacles, and point obstacles)—specifically, a safety range is demarcated outward from the boundary of each obstacle. Among these, the buffer zone for artificial obstacles (e.g., no-navigation zones, aquaculture zones) corresponds to a 1-nautical-mile safe distance, which aligns with the obstacle avoidance standards recommended in the textbook Route Design [28]. This design can effectively reduce route risks and support the validity of the proposed method.

These factors are then incorporated into route planning to assist ships in avoiding obstacles and staying clear of dangerous areas.

3.1.2. Dynamic Environmental Fields

Waves significantly impact a vessel’s stability, maneuverability, and navigational comfort, making them a core element of the dynamic environment. The European Centre for Medium-Range Weather Forecasts (ECMWF) is renowned for its high-resolution and accurate data, as is illustrated in Figure 4. Since its inception in 1979, ECMWF has established a leading position in the field of numerical weather prediction, attracting widespread attention from meteorological researchers and related industries. This study utilizes wave forecast data provided by ECMWF, which covers forecast lead times from 0 to 144 h with a time resolution of 3 h [29]. The primary elements of the wave forecast data include:

(1)

Significant Wave Height (SWH): Also known as the significant wave amplitude, it refers to the average of the highest one-third of wave heights observed, used to describe the overall sea state.

(2)

Mean Wave Direction: This indicates the average direction of wave propagation and has a significant impact on the vessel’s heading choice and stability.

(3)

Wave Peak Period: This refers to the time interval between two consecutive wave peaks and has a notable effect on the vessel’s pitching and rolling characteristics.

By analyzing ECMWF’s wave forecast data, it is possible to accurately predict wave conditions during navigation, thereby optimizing the route, improving both safety and efficiency, and ensuring that the vessel completes its journey safely and efficiently under various sea conditions.

3.2. Ship Motion Model

The ship motion model, particularly the model describing ship motion in waves, is a critical parameter for evaluating navigation time and route safety. This study primarily considers ship stall due to waves and the operational safety limits under wave conditions.

3.2.1. Stall Model for Ships Navigating in Waves

When navigating at sea, a ship’s speed typically decreases compared to its speed in still water, even with constant engine power, due to wind and wave disturbances. This phenomenon is commonly referred to as “stall.” The Khokhlov formula [30] is a stall model that comprehensively takes into account factors such as wave height, wave direction, ship dimensions, and ship tonnage. The calculation formula for this model is as follows:

$$v=v_0-\left(0.745\cdot H_S-0.245\cdot \theta \cdot H_S\right)\cdot \left(1.0-1.35\cdot {10}^{-6}\cdot D\cdot v_0\right)$$

(2)

In this formula, θ represents the relative angle between the waves and the ship (in radians), and D denotes the ship’s tonnage.

3.2.2. Safety Limitations for Ships Navigating in Waves

(1)

Parametric Rolling

Parametric rolling refers to the phenomenon where a ship loses control of its propulsion system while navigating in waves, causing the ship to move with the waves. This operation can result in a risk of capsizing due to sudden changes in heading and unexpected large tilts. According to the relevant regulations of the International Maritime Organization (IMO) [31], where α denotes the course angle, parametric rolling may occur when a ship navigating in wind and waves meets the following conditions:

$$\left\{\begin{array}{c}135°\le \alpha \le 225°\\ v\ge {\displaystyle \frac{1.8\sqrt{L_{ship}}}{\cos \left(180°-\alpha \right)}}\end{array}\right.$$

(3)

(2)

Maximum Wave Height Limit

In wave conditions, a ship’s safe navigation is limited by the maximum wave height. According to relevant regulations from the International Maritime Organization (IMO), the maximum wave height must be considered during ship design and operation to ensure safety under extreme sea conditions. The maximum wave height limit not only relates to the ship’s stability and structural strength but also affects its seaworthiness and the safety of the crew. This study utilizes forecast data provided by ECMWF to assess the maximum wave height in the navigable area. When the predicted wave height exceeds the maximum wave height allowed by the ship’s design and operational limits, the ship is restricted from entering these areas to reduce navigational risks.

3.3. Dynamic Update of Time Weights in the Route Network

In the ship experience route network generated by the method in Section 2.2, the weight of a route is calculated based on the route length. However, to further optimize the navigation time and ensure the shortest travel time for the planned route, a network based solely on distance weights is insufficient. Therefore, it is necessary to convert the existing distance-weighted route network into a time-weighted route network. The construction method of a time-weighted route network typically involves calculating the time weight by dividing the route distance by the ship’s speed. However, the marine environment not only affects the ship’s speed but it may also render certain routes impassable at specific times. To address this, it is necessary to integrate the ship’s motion model and comprehensively consider the impact of dynamic marine environmental factors on navigation to adjust the ship’s speed, thus updating the time weights in the route network in real time. The key steps are as follows:

(1)

Marine Environment Data Acquisition

ECMWF data is provided in a gridded format. To obtain more accurate marine environmental forecast values for a specific location, spatial and temporal linear interpolation is required. First, spatial linear interpolation is performed using data from the four neighboring grid points surrounding the target location to obtain forecast values for different time points at each location. Then, based on these spatial interpolation results, further linear interpolation is performed with respect to time to obtain the final forecast value at the specified location and time.

➀ Temporal Linear Interpolation

For a given location (x, y), the forecast value P(x, y, t) at time t is sought. Since ECMWF provides forecast data as discrete grid data corresponding to times t₀ and t₁, temporal linear interpolation can be performed as follows:

$$P\left(x,y,t\right)=P\left(x,y,t_0\right)+{\displaystyle \frac{t-t_0}{t_1-t_0}}\left(P\left(x,y,t_1\right)-P\left(x,y,t_0\right)\right)$$

(4)

In the formula, P(x, y, t₀) represents the forecast value at location (x, y) at time t₀, and P(x, y, t₁) represents the forecast value at location (x, y) at time t₁.

➁ SpatialLinearInterpolation

After temporal interpolation is completed, spatial interpolation is needed for the spatial coordinates. The forecast data at the four grid points closest to the target location (x, y) at time t are P₁(x₁, y₁, t), P₂(x₂, y₂, t), P₃(x₃, y₃, t), and P₄(x₄, y₄, t). Next, bilinear interpolation can be used to calculate the value of the target location P(x, y, t). First, linear interpolation is performed in the x-direction to obtain the intermediate values P₁′ and P₂′:

$$\left\{\begin{array}{c}{P}_1^{\prime }=P_1\left(x_1,y_1,t\right)+{\displaystyle \frac{x-x_1}{x_2-x_1}}\left(P_2\left(x_2,y_2,t\right)-P_1\left(x_1,y_1,t\right)\right)\\ P_2^{\prime }=P_3\left(x_3,y_3,t\right)+{\displaystyle \frac{x-x_3}{x_4-x_3}}\left(P_4\left(x_4,y_4,t\right)-P_3\left(x_3,y_3,t\right)\right)\end{array}\right.$$

(5)

Then, linear interpolation is performed in the y-direction to obtain the final interpolated result P(x, y, t):

$$P\left(x,y,t\right)=P_1^{\prime }+{\displaystyle \frac{y-y_1}{y_3-y_1}}\left(P_2^{\prime }-P_1^{\prime }\right)$$

(6)

(2)

Stall Analysis for Ship Navigation in Wind and Waves

Using the method in (1), the marine environmental data for a specific time and location can be calculated. Next, based on Equation (2), the ship’s speed under these environmental conditions can be determined, and it can be assessed whether the area is navigable. As an example, consider the position at 25° N latitude and 120° E longitude at 00:00 on 17 October 2024, where the wave height is 3.5 m and the wave direction is 30 degrees. When the ship’s main engine speed is 18 knots, the ship’s speed variation radar chart for headings from 0° to 360° is shown in Figure 5. From the radar chart, it can be observed that the ship’s speed is significantly influenced by marine environmental factors such as wave height and wave direction.

(3)

Segmented Solution for Route Weight Calculation

The route length in the route network is often uncertain. Therefore, to improve the accuracy of voyage estimation, the route needs to be divided into segments. For each segment, the solution process is as follows: First, starting from the node’s starting position, the ship’s speed at the current time is calculated based on Equation (2). Then, the ship’s position one hour later is computed based on the current speed, and the environmental data is updated to recalculate the ship’s speed at the new time. Finally, based on the updated speed, the ship’s position is calculated hour by hour towards the destination, progressively solving for the navigation time until the endpoint of the segment is reached. Through this segmented solution approach, the route network can dynamically update the time weights, flexibly adapting to changes in sea conditions, thereby providing a more accurate navigation time estimate and optimizing route planning.

4. Optimal Time Weather Route Generation Based on an Improved Time-Variant A* Algorithm

This paper proposes an improved time-variant A* algorithm that dynamically calculates and adjusts route planning in real-time based on a time-weighted route network, thereby optimizing the navigation strategy and providing safe and efficient route planning for ships in complex marine environments.

(1)

Dynamic Time Cost Calculation

In the time-variant A* algorithm, the cost of each node is composed of time weights. The time weight T_ij(t) for the route from node i to node j changes over time t and is calculated using the following formula:

$$T_{ij}\left(t\right)={\displaystyle \frac{d_{ij}}{v_{ij}\left(t\right)}}$$

(7)

where d_ij is the spatial distance of the route segment from node i to node j, and v_ij(t) is the actual ship speed under the current marine environment, which can be solved using Equation (2).

Once the time cost T_ij(t) for each route segment is determined, the overall cost function for the route can be expressed as:

$$f\left(i\right)=g(i)+h(i)$$

(8)

where g(i) is the actual accumulated time cost from the starting point to the current node i, and h(i) is the estimated time cost from node i to the destination node, determined by the time estimate influenced by the current environmental conditions. The heuristic function h(i) can be dynamically adjusted based on the current time and marine environmental data.

(2)

Real-Time Route Optimization with the Time-Variant A* Algorithm

The real-time route optimization using the time-variant A* algorithm selects the optimal path through real-time iteration and dynamically optimizes the route planning based on the latest environmental data. During each search, the algorithm considers the impact of environmental factors on ship speed through the cost function, thereby adjusting the route to ensure that the mission is completed in the shortest time, even under complex weather conditions. In the iteration process, the time-variant A* algorithm updates the time cost for each route segment and reassesses the path accordingly. This not only optimizes the cost of the current path but also estimates the future path costs, particularly in response to potential weather changes. For example, if the area ahead is a large wave zone, the algorithm will adjust the heading to avoid unfavorable regions, thereby reducing the navigation time. The real-time optimization capability of the time-variant A* algorithm enables it to cope with sudden weather changes and instantly correct the route based on new weather conditions, enhancing the flexibility and safety of route planning.

5. Case Study

5.1. Background

To validate the effectiveness of the proposed method, this paper uses the ship weather routing software SIMROUTE [23] as a comparison method for the experiments, with grid sizes of 0.5° and 0.1°. The time window for the experimental data is set as the seven-day forecast data from the European Centre for Medium-Range Weather Forecasts (ECMWF) for 15 October 2024. The identification of intersections between routes and restricted areas is implemented by the GIS topological intersection method. Figure 6 shows a schematic of the wave height and wave direction forecast data in the longitude range of 100–150° and latitude range of 0–50°. The experimental ship parameters are set as follows: length 135 m, width 20 m, deadweight tonnage 4000 tons, with a speed of 18 knots. The maximum allowable wave height limit for the ship is set to 5 m. The experimental environment includes the following: operating system (Windows 11), processor (Intel Core i9-12,900 H, 2.50 GHz), memory (64 GB RAM), and programming language (Python 3.8.10). The experiment is divided into three parts: Case 1 involves a route from Busan, South Korea, to Tianjin, China, aimed at testing the planning capability for ocean routes; Case 2 involves a route from Weihai to Ningde, aimed at testing the capability for coastal navigation; Case 3 is a comparative experiment on multi-group meteorological route planning.

5.2. Case 1: Ship Ocean Route Planning in Meteorological Conditions

Case 1 is a ship ocean route planning experiment, with the voyage from Busan, South Korea, to Tianjin, China, passing through the Sea of Japan, the Yellow Sea, and the Bohai Sea. The sea conditions along this route are relatively complex, with many islands and reefs, narrow channels, and limited water depth. The experiment sets the ship to depart from the starting position (133.0° E, 36.0° N) at 00:00 on 15 October 2024, with the destination at (118.5° E, 38.5° N). Both SIMROUTE and the proposed method are used to generate meteorological routes. The generated route schematic is shown in Figure 7, with an enlarged display of some complex areas in Figure 8. In this case, the route generated by SIMROUTE using a 0.5° grid is labeled Route1, the route generated using a 0.1° grid is labeled Route2, and the route generated by the proposed method is labeled Route3. The comparison results of navigation time, route length, number of intersections with restricted areas, and computation time are shown in

Table 1. Comparison Results of Weather Routes in Case 1.

Method	Navigation Time (s)	Route Length (nm)	Number of Intersections with Restricted Area	Computation Time (s)
SIMROUTE (0.5°)	185,040.6	891.64	12	20.3
SIMROUTE (0.1°)	185,586.2	897.61	8	528.89
Proposed Method	175,987.6	845.15	0	35.7

.

From Table 1, it can be seen that the meteorological route generated by the proposed method does not intersect with any restricted areas, whereas the meteorological routes generated by SIMROUTE, both using the 0.1° and 0.5° grids, intersect multiple times with restricted areas, posing potential safety risks. From Figure 8a, it can be observed that in the southern waters of South Korea, in the island and reef zone, both Route1 and Route2 generated by SIMROUTE intersect several times with restricted areas. This is because, despite a 0.1° grid being relatively fine, the grid scale is still too large relative to small islands and reefs. In contrast, the proposed method uses an adaptive quadtree grid, which automatically refines the area when a restricted zone is detected. Additionally, the route network in the proposed method is based on historical ship trajectories, typically constructing the network only within navigable waters, which effectively avoids passing through restricted areas.

From Table 1, it can be seen that the route generated by the proposed method, compared to SIMROUTE, has an average route length reduction of 5.8% and an average navigation time reduction of 5.1%, achieving effective route optimization. By enlarging Figure 8a,b, it can be observed that, whether using Route1 generated with a 0.5° grid or Route2 generated with a 0.1° grid, both follow a regular grid search method and navigate along neighboring points. The trajectories and route lengths of both are almost identical. In fact, as seen in Table 1, because Route1 uses a larger grid size, it overlooks some obstacle-avoidance details, which results in a shorter route length. In contrast, the proposed method, based on historical ship trajectories, allows for two grid points, even if they are far apart, to be directly connected to form a route. As shown in Figure 8b, this method directly connects from the starting point to the southern waters of South Korea. Therefore, the proposed method overcomes some limitations of the A* algorithm’s local greedy strategy, effectively shortening both the route length and navigation time.

From Table 1, it is also evident that the proposed method is relatively faster in computation. Although its computation speed is 43.1% slower compared to SIMROUTE when using a 0.5° grid for rough planning, the proposed method is 14.8 times faster compared to SIMROUTE using a 0.1° grid for finer-grained planning.

In conclusion, the experimental results show that the proposed method can effectively guide ships around restricted areas, ensuring route safety while reducing navigation time. It is particularly suitable for crossing complex marine areas, offering the advantages of navigability, safety, and time-efficiency.

5.3. Case 2: Ship Coastal Navigation in Meteorological Route Planning

Case 2 focuses on meteorological route planning for coastal navigation, with the voyage from Weihai to Ningde. The latitude span is relatively large, and during the voyage, the varying sea conditions in different areas need to be considered. The route primarily follows the coastline. The experiment sets the ship to depart from the starting position (122.21° E, 37.5° N) at 00:00 on 18 October 2024, with the destination at (120.5° E, 26.5° N). Both SIMROUTE and the proposed method are used to generate meteorological routes. The generated route schematic is shown in Figure 9, with an enlarged display of some areas in Figure 10. In this case, the route generated by SIMROUTE using a 0.5° grid is labeled Route1, the route generated using a 0.1° grid is labeled Route2, and the route generated by the proposed method is labeled Route3. The comparison results for navigation time, route length, number of intersections with restricted areas, and computation time are shown in

Table 2. Comparison Results of Weather Routes in Case 2.

Method	Navigation Time (s)	Route Length (nm)	Number of Intersections with Restricted Area	Computation Time (s)
SIMROUTE (0.5°)	148,913.2	730.37	7	10.6
SIMROUTE (0.1°)	147,467.6	729.45	6	140.34
Proposed Method	144,976.9	710.21	0	39.29

.

From Table 2, it can be seen that the meteorological route generated by the proposed method does not intersect with any restricted areas, whereas the meteorological routes generated by SIMROUTE, both using the 0.1° and 0.5° grids, intersect multiple times with restricted areas, posing potential safety risks. From Figure 10a, it can be observed that in the waters near Zhoushan, both Route1 and Route2 generated by SIMROUTE intersect several times with restricted areas, whereas the proposed method directly navigates around the restricted areas in safe waters. This is because, even though SIMROUTE uses a 0.1° grid, the grid scale is still too large relative to small islands and reefs, causing the route to pass through smaller islands. In contrast, the proposed method uses an adaptive quadtree grid, which can adaptively subdivide and refine the planning granularity when encountering restricted areas.

From

Table 3. Start and End Positions of Experimental Routes.

Experiment No.	Start Longitude (°)	Start Latitude (°)	End Longitude (°)	End Latitude (°)
1	138.0	38.0	122.5	31.0
2	125.5	35.0	123.5	30.0
3	124.0	38.5	121.6	37.6
4	121.76	30.43	123.8	29.8
5	122.4	28.26	112.2	20.3

, it can be seen that the route generated by the proposed method, compared to SIMROUTE, has an average route length reduction of 2.5% and an average navigation time reduction of 2.6%. Overall, although the optimization of the route is not large, Figure 9 shows that many of the improvements in route length by SIMROUTE do not adhere to navigation safety rules. For example, Figure 10a shows that when passing through the Zhoushan area, SIMROUTE’s route directly crosses the island and reef area, which shortens the route length but compromises navigation safety. In contrast, the proposed method strictly navigates around the island and reef area. Similarly, as shown in Figure 10b, during coastal navigation, SIMROUTE’s route runs close to the mainland edge to shorten the distance, while the proposed method maintains an appropriate offshore distance.

From Table 3, it is also evident that the proposed method is relatively faster in computation. Although the computation time is 2.7 times longer than SIMROUTE when using a 0.5° grid for rough planning, the proposed method is 3.57 times faster when compared to SIMROUTE using a 0.1° grid for finer-grained planning. Due to the higher density of restricted areas in coastal waters, the adaptive grids generated by the proposed method are denser, resulting in slight reductions in computational speed compared to the ocean routes in Case 1. However, for longer ocean routes, the improvement in computation speed will be more significant.

In conclusion, the experimental results show that the proposed method can fully utilize hydro-meteorological factors and effectively guide ships around restricted areas, resulting in better meteorological route planning for coastal navigation.

5.4. Case 3: Comparative Experiments on Multi-Group Meteorological Route Planning

To further verify the stability and universality of the proposed method under different navigation scenarios and eliminate the randomness of single-route experiments, 5 groups of comparative experiments on meteorological route planning were designed. The experiments cover typical complex sea areas of the Northwest Pacific Ocean and the eastern coast of China, aiming to comprehensively test the safety performance, optimization effect, and computational efficiency of the method under different route distances and sea area types.

The start and end points of each group of experiments are shown in Table 3. The routes cover various typical sea environments, such as island reef areas (e.g., around the Zhoushan Archipelago), open sea areas (central Yellow Sea), and straits, with a length range of 300–1200 nautical miles. They include both medium-short-distance coastal routes and long-distance cross-sea routes, simulating different task requirements in actual shipping. The experimental parameters are consistent with those in Section 5.1, and the comparison method remains SIMROUTE (0.5° grid, 0.1° grid). The comparison results of navigation time, route length, number of intersections with obstacle areas, and computation time are shown in

Table 4. Comparative Results of Meteorological Routes in Case 3.

Experiment No.	Method	Navigation Time (s)	Route Length (nm)	Number of Intersections with Restricted Area	Computation Time (s)
1	SIMROUTE (0.5°)	194,124.33	954.49	9	10.89
	SIMROUTE (0.1°)	194,069.78	954.92	5	286.94
	Proposed Method	188,192.70	923.25	0	48.24
2	SIMROUTE (0.5°)	68,496.52	335.50	1	2.16
	SIMROUTE (0.1°)	68,456.76	335.63	1	48.97
	Proposed Method	68,217.36	339.28	0	9.97
3	SIMROUTE (0.5°)	31,508.52	156.81	3	0.89
	SIMROUTE (0.1°)	27,991.91	139.42	2	10.82
	Proposed Method	27,765.79	138.22	0	1.07
4	SIMROUTE (0.5°)	30,492.63	148.68	5	0.66
	SIMROUTE (0.1°)	30,384.72	230.21	2	67.32
	Proposed Method	23,636.13	115.42	0	16.78
5	SIMROUTE (0.5°)	167,865.08	836.51	14	6.08
	SIMROUTE (0.1°)	157,774.46	786.97	9	112.93
	Proposed Method	149,973.57	748.10	0	14.37

.

In the 5 groups of experiments, the number of intersections with restricted areas of the proposed method is 0, completely avoiding risk areas such as reefs and no-navigation zones; while SIMROUTE has an average of 6.4 intersections when using a 0.5° grid and 3.8 intersections when using a 0.1° grid, with the number increasing significantly in the long-distance route experiments. This verifies that the method proposed in this paper can effectively adapt to coastal and island-rich waters and plan safe and feasible routes based on historical trajectories.

Compared with SIMROUTE, the average route length of the proposed method is reduced by 6.8–7.4%. Specifically, the average route length of SIMROUTE is reduced from 2431.99 nautical miles to 2264.27 nautical miles when using a 0.5° grid, and from 2447.15 nautical miles to 2264.27 nautical miles when using a 0.1° grid; the average navigation time is reduced by 4.3–7.1%. This is because the proposed method can construct direct routes based on historical trajectories, breaking through the adjacent point search limitation of regular grids. Meanwhile, the method can dynamically avoid severe weather areas, thus achieving better route planning results on the premise of ensuring safety.

It can be seen from the data of 10 groups of multi-scenario experiments that the average computation time of the proposed method is 90.43 s, which is only 17.1% of that of SIMROUTE when using a 0.1° grid (526.98 s); although it is longer than that of SIMROUTE when using a 0.5° grid (20.68 s), it can simultaneously achieve the safety performance at the 0.1° grid level and the computational efficiency close to that at the 0.5° grid level. This is attributed to two factors: first, the on-demand refinement feature of the adaptive grid, which only refines the grid in complex sea areas, greatly reducing redundant computations; second, the network simplification effect brought by trajectory clustering, which effectively reduces the algorithm search space by eliminating invalid connections.

The 5 groups of multi-scenario experiments further verify that the proposed method can stably achieve the coordinated improvement of safety, optimization, and efficiency under different route types and sea area environments.

6. Conclusions

In the field of meteorological route planning, ensuring both safety and efficiency in coastal and island-rich waters has always been a challenge. Developing a method that effectively addresses the complexities of the marine environment and improves route planning safety and efficiency is of significant economic and safety importance for the maritime transport industry. This study proposes an efficient meteorological route planning method for coastal and island-rich waters, guided by ship trajectory big data. The core innovation lies in constructing a ship’s empirical route network and dynamically integrating oceanic environmental factors into the route planning process, thereby enhancing both the safety and efficiency of the routes. The method is validated through theoretical analysis and case studies, and a comparison with the existing SIMROUTE method leads to the following key conclusions:

(1)

The proposed method significantly improves the safety of weather routing in coastal and island-rich waters. Experimental results show that the weather routes planned using the proposed method do not intersect with any restricted areas, whereas the comparative methods frequently intersect with small islands and reefs.

(2)

The proposed method effectively optimizes computational efficiency. Experimental results demonstrate that, compared to existing methods, the proposed method improves computational efficiency by at least three times in terms of high-precision route planning speed.

(3)

The proposed method makes full use of meteorological factors, enabling the effective planning of navigable, safe, and time-efficient routes for coastal and island-rich waters. Experimental results show that the routes planned with this method are shorter in terms of both route length and sailing time.

In conclusion, the meteorological route planning method proposed in this study demonstrates significant advantages in enhancing navigation safety and efficiency, with particular potential for application in coastal and island-rich waters. These findings not only advance the development of route planning technologies but also provide new insights and tools for practical operations in the maritime transport industry. However, the proposed method relies on substantial volumes of high-quality ship trajectory data to support the construction of the empirical route network for meteorological route planning. Specifically, if there is a complete lack of available trajectory data in the target sea area (e.g., undeveloped remote coastal areas), it will be impossible to generate initial network nodes and connection relationships, rendering the method inapplicable for route planning. Even when trajectory data exists, its volume directly affects the calculation accuracy of route weights—the more sufficient the data, the more realistic the weight assignment. Currently, the quantitative evaluation of the relationship between data volume, weight accuracy, and planning effectiveness has not been completed, and how to optimize the quality of meteorological route planning under limited data resources remains a key issue to be addressed in future research.