Short-Term Drifter Position Estimation Based on Ship-Generated Waves for Search and Rescue

Article information

J. Ocean Eng. Technol. 2026;40(1):77-84

Publication date (electronic) : 2025 December 22

doi : https://doi.org/10.26748/KSOE.2025.048

Yang Zhao¹, Jung-Ho Kang¹, Kyung-Chang Lee²

¹Graduate Student, Department of Intelligent Robot Engineering, Pukyong National University, Busan, Republic of Korea

²Professor, Department of Intelligent Robot Engineering, Pukyong National University, Busan, Republic of Korea

Corresponding author Kyung-Chang Lee: +82-51-629-6332, gclee@pknu.ac.kr

Received 2025 August 26; Revised 2025 October 20; Accepted 2025 December 8.

Abstract

1. Introduction

When a person falls overboard at sea, the critical rescue period is typically within 10 min, which is extremely short, and the survival time decreases significantly depending on the water temperature (Piantadosi, 2003; Huang, 2014). Therefore, accurately estimating the initial drifter position in man overboard incidents, which serves as the starting point for search operations, is critical for improving both rescue efficiency and survival probability. In maritime environments, where external factors such as currents, waves, and weather interact in complex ways, the drift position of a person can change rapidly, causing the search area to expand exponentially (Wu et al., 2023; Zhou et al., 2020; Allen & Plourde, 1999; Allen, 2005). Owing to these characteristics, maritime Search and Rescue (SAR) operations are essentially a race against time, making it crucial to estimate a person’s drift position accurately and swiftly (Kim et al., 2023; Brushett et al., 2017; Breivik et al., 2011; Mou et al., 2021).

Traditionally, the Leeway model has been widely used to estimate the drift at sea (Kratzke et al., 2010; Daniel et al., 2003; Guard, 2009). This model effectively estimates the medium- to long-term drift trajectories of persons overboard or floating objects based on wind forces and is applicable under various conditions (Wu et al., 2023; Brushett et al., 2014; Huang et al., 2014; Meng et al., 2021; Kasyk et al., 2021; Tu et al., 2021; Feraru et al., 2020). To improve the accuracy of short-term drift position estimation, Feraru et al. (2020) assumed that the crosswind drift component is negligible because of the shape of the human body and therefore considered only the downwind component. However, because the Leeway model was originally designed for simulations spanning several hours or more, it has inherent limitations in minimizing short-term position errors that occur within minutes after an accident.

Recently, data-driven approaches, such as machine learning, have been applied to marine drift prediction, showing improved accuracy over traditional numerical models (Nam et al., 2020; Lin et al., 2024). However, as these methods rely heavily on the density and temporal resolution of historical data, their applicability remains limited to minute-scale localization immediately after a man overboard incident.

For example, as shown in Fig. 1, if ocean currents or ship-generated waves exist in a direction perpendicular to the prevailing wind, the actual position of a person overboard may deviate from that estimated using the Leeway model. This deviation can cause errors in estimating the initial position when determining the starting point of the search.

Fig. 1

Initial drifter position estimation in man overboard incidents

Such errors directly affect the success or failure of the rescue operations. However, in actual SAR scenarios, the possibilities for compensation are limited.

In this study, we aimed to overcome the limitations of the Leeway model for short-term drift estimation by explicitly incorporating the effects of ship-generated waves. In maritime accidents that occur when a ship is in operation, the initial drift position is significantly influenced not only by ocean currents and wind speed but also by the ship-generated waves created during navigation (Newman, 2018). As shown in Fig. 2, these waves form nonlinear wave height distributions that extend for tens of meters across the sea surface around the hull. Their structures and magnitudes vary depending on the shape and speed of the ship. Consequently, ship-generated waves can have a decisive impact on short-term drift position estimation in man-overboard incidents during the first few minutes of an accident.

Fig. 2

Ship-generated waves during sailing

To this end, in this study we modeled ship-generated waves using Kelvin’s wave theory and constructed a wave-induced velocity field based on the potential flow analysis. The wave-induced velocities calculated from this field were then integrated into the conventional Leeway model to propose an improved drift position estimation method. Using this approach, the positional accuracy of the initial search point immediately after an overboard incident can be enhanced.

The remainder of this paper is organized as follows. Section 2 introduces the conventional Leeway model, and Section 3 presents the proposed short-term drift estimation model that accounts for the effects of ship-generated waves. For this purpose, the wave height distribution of the ship-generated waves was first analytically modeled, and a gridded sea surface velocity field was constructed. The induced velocities derived from this field were calculated and integrated into the Leeway model. In Section 4, the effectiveness of the proposed model is verified through numerical simulations, and the improvement in the drift position estimation accuracy is evaluated when incorporating the wave-induced correction term. Finally, Section 5 discusses conclusions and directions for future research.

2. Leeway based Drift Position Estimation Model

In maritime rescue, the Leeway model, which is widely used to estimate the drift of persons overboard (Allen & Plourde, 1999), calculates the Leeway drift vector (V→_L), which represents the direction and velocity of a floating object’s movement under the influence of wind (W), as shown in Fig. 3. As expressed in Eq. (1), the vector V→_L is composed of the sum of two components: the Downwind Leeway Component (L→_DWL), which corresponds to the movement in the direction of the wind, and the Crosswind Leeway Component (L→_±CWL), which corresponds to the lateral movement to the left or right relative to the wind direction. Each component(L→_DWL, L→_±CWL) is characterized by three drift parameters, a, b, and ɛ, which are determined under various sea conditions. Parameter a represents the slope relating wind speed to the drift component, b is the intercept corresponding to the residual drift under calm or zero-wind conditions, and ɛ denotes the random residual that accounts for measurement scatter.

Fig. 3

Diagram of the Leeway model showing the wind-induced drift vector

The Leeway model demonstrates effective performance in estimating drift over several hours because it is based on statistical drift parameters derived from long-term observational data under various sea conditions. In addition, Wu et al. (2023) and Tu et al. (2021) improved the accuracy of the long-term drift position estimation by adding a current-induced component (V→_Flow) to the wind-based drift velocity (V→_L) calculated using the Leeway model. However, for short-term drift estimation within a few minutes of a maritime accident, the Leeway model faces critical limitations. This is because, in cases where a person falls overboard a ship, not only wind and ocean currents but also ship-generated wave-induced flows exert a significant influence on the movement of the person.

Given that the Leeway model does not consider the effects of ship-generated wave-induced flows, discrepancies of several tens of meters can occur between the actual and estimated positions of the model. Accordingly, this study proposes an integrated estimation model that improves the short-term accuracy of drift position estimation by adding wave-induced velocity as a correction term to the Leeway model, as formulated in Eq. (2).

3. Ship-Generated Waves Based the Drifter Position Estimation Model

This section presents a drifter position estimation model that improves the accuracy by incorporating the nonlinear effects of ship-generated waves. Ship-generated waves were modeled, and a gridded wave-induced-velocity field was generated. The velocity components derived from this field were subsequently integrated into the Leeway model.

3.1 Ship-Generated Wave Modeling

When a ship sails on the sea surface, the interaction between the hull and fluid generates ship-generated waves on the free surface (Min et al., 2019), forming nonlinear wave patterns and wave-height distributions. The characteristics of ship-generated waves vary significantly depending on the physical conditions, such as the shape, size, and speed of the ship.

Representative approaches to modeling ship-generated waves include numerical methods based on computational fluid dynamics (CFD) and analytical methods based on Kelvin’s ship-wave model. The CFD approach numerically solves the Navier–Stokes equations and combines them with free-surface tracking algorithms to accurately reproduce the dynamic response of the water surface around the hull. Although this method can capture complex physical phenomena in detail and is suitable for high-precision simulations, it has the drawbacks of a long computational time and the need for high-performance computational resources.

In contrast, the Kelvin ship-wave model is based on the wavemaker concept, linear potential flow theory, and stationary phase method, and analytically computes the wave structure of the wake generated by a ship. This model can estimate the transverse and divergent waves that form distinct V-shaped patterns depending on the ship speed, shape, and direction of motion. Owing to its high computational efficiency, it is well suited for real-time estimation and rapid analysis.

In this study, to compare and analyze the differences in ship wave characteristics across the two modeling methods, simulations were conducted by applying both CFD and the Kelvin model to geometrically identical hulls at multiple scales. The simulation environments for the CFD and Kelvin models are presented in Table 1.

Table 1

Comparison of simulation environments for the CFD and Kelvin models

Fig. 4 shows the results of the CFD-based simulations performed under a ship speed condition of 7.6 kn (3.91 m/s), using ship models of the same hull form but with adjusted draft (d), beam (b), and length (l) to generate four geometrically scaled models, ranging from a small ship (Fig. 4(a)) to a large ship (Fig. 4(c)). The results confirmed that even under identical speed conditions, the characteristics of ship-generated waves varied depending on the ship scale.

Fig. 4

Results of CFD-based modeling of ship-generated waves

Fig. 5 shows the ship-generated wave modeling results calculated using the Kelvin model under the same hull form, scale conditions, and ship speed. A comparative analysis of the two modeling approaches indicated that as the ship scale increased, the Kelvin model and CFD simulations exhibited similar trends in wave patterns and structural features.

Fig. 5

Results of Kelvin-based modeling of ship-generated waves

However, the Kelvin model is generally valid when the ship travels in a steady and rectilinear motion, the wave steepness is small, and the water is sufficiently deep. In addition, the model considers only simple geometric parameters, such as ship speed, length, beam, and draft, while neglecting complex hydrodynamic effects, including viscous dissipation and strong nonlinearity. Consequently, under rough sea conditions, during ship acceleration or turning maneuvers, or in cases of multihull interference, the Kelvin wake pattern may deviate from the actual wavefield. Consequently, the wave energy decay, wave crest shape, and wake angle variation can be predicted more mildly than in reality, indicating that the model has limitations in terms of local wave height prediction accuracy compared with CFD-based analysis.

Nevertheless, the Kelvin model offers the advantages of high computational efficiency and ease of implementation, enabling the rapid assessment of the overall wave distribution characteristics generated during ship navigation without the need for high-performance computing resources. Therefore, in this study, the Kelvin model was used to model the ship-generated waves.

3.2 Gridding and Wave-Induced Velocity Field Calculation

In ship-generated wave modeling, the wave height distribution of the sea surface around a ship is provided as continuous surface elevation data. However, to integrate this information into the Leeway model, the continuous data must be discretized into a grid, and the physical velocity vector at each grid point must be calculated. In this study, a uniform Cartesian sampling grid (Aldroubi & Gröchenig, 2001) was applied by dividing the region affected by ship-generated waves into regular grids at 0.05-m intervals along the x- and y-axes. This approach increases the computational efficiency and enables the calculation of the physical velocity vectors at each grid point.

At each discretized grid point, the wave heights calculated by the Kelvin model were assigned, and based on this information, the local wave-induced velocity vector was computed using linear potential flow theory. Linear potential flow theory assumes that the fluid is inviscid, irrotational, and incompressible, and expresses the fluid velocity vector as the spatial gradient of the potential function, as shown in Eq. (3); here V→(x,y) denotes the horizontal free-surface velocity (m/s), φ is the velocity potential (m²/s), and ∇ is the horizontal gradient operator (m⁻¹); x and y are horizontal coordinates in meters.

(3) V→(x,y)=∇φ

However, directly calculating the potential function φ is inefficient in terms of computational performance. Therefore, in this study, under the assumption of linear small-amplitude waves, an approximate relationship with the wave height function η (m) on the free surface was employed, as expressed in Eq. (4). The potential function on the free surface can be approximated as here, g = 9.81 m/s² is the gravitational acceleration and Ω (rad/s) is the angular frequency of the wave, which is determined by the dispersion relation of gravity waves.

(4) φ≈gωη

Assuming that the phase velocity of the Kelvin wave matches the speed of the ship, the velocity components at each grid point can be calculated based on this relationship, as expressed in Eq. (5).

(5) V→(x,y)=∇∅=∇gωη

Through this process, the wave-induced velocity vector at each grid point was calculated, and the velocity field for the entire ship-generated wave region was constructed. In the practical application stage, the wave-induced velocity vector V→_Shipwave (m/s) at a specific location was determined using the weighted average of the velocity vectors ( V1→,V2→,V3→,V4→) from the four surrounding grid points, as shown in Fig. 6. This result was used as the input value for the drift position estimation model.

Fig. 6

Ship-generated wave-induced velocity vector derived from weighted averaging of grid point velocities

4. Experiments and Performance Evaluation

In this section, numerical simulations were conducted by integrating the Kelvin-wave-induced velocity into the Leeway model as a correction term. Comparative experiments were conducted by analyzing the drift position estimation results of the conventional leeway model and the proposed model with the wave-induced velocity incorporated under identical environmental conditions.

The Leeway model consists of downwind and crosswind components, with the crosswind direction randomly determined as left or right with equal probabilities. In single-target simulations, this randomness can result in a trajectory bias. Therefore, in this study, a Monte Carlo simulation was performed with 100 targets, and the crosswind direction was reassigned every 5 s. The coordinates at each time step were then averaged to derive a representative drift trajectory that minimized the random deviations.

In addition, assuming that the clothing condition and posture of the person overboard were uncertain, the “unknown state” parameters (a_DWL = 0.0096, b_DWL = 0, ɛ_DWL = 0.12, a_±CWL = 0.0054, b_±CWL = 0, ɛ_±CWL = 0.094) were applied.

The Leeway model consists of downwind and crosswind components, with the crosswind direction randomly determined to the left or right with equal probability. In single-target simulations, this randomness can cause trajectory bias; therefore, in this study, a Monte Carlo simulation was performed with 100 targets, reassigning the crosswind direction every 5 s. The coordinates at each time step were then averaged to derive a representative drift trajectory that minimizes random deviations.

Fig. 7 shows an example in which the initial drift position is set to the beam position corresponding to the ship centerline. The total simulation time was 10 min, the coordinates were updated every 5 s, and the crosswind direction was randomly reassigned at each interval. For trajectory visualization, the positions were marked at 2-min intervals to allow for the intuitive observation of drift movement over time.

Fig. 7

Initial drift positions in small and large ship simulations

Fig. 8 shows a comparison of the mean drift trajectories between the conventional Leeway model and the proposed model under both small and large ship conditions. The results indicate that incorporating the wave-induced velocity resulted in clear differences in the final drift positions in both cases. Under the small-ship condition, the drift moved 6.7 m farther along the Y-axis toward the ship’s starboard side and decreased by 15.5 m along the X-axis. Under the large-ship condition, the drift moved 9.5 m farther along the Y-axis and decreased by 32.7 m along the X-axis. These results indicate that under identical environmental conditions, ship-generated waves produced by large ships exert a stronger disruptive influence on the drift behavior of persons overboard.

Fig. 8

Comparison of mean drift trajectories under small and large ship conditions between the Leeway model and the Leeway+ ship-wave model

In this study, simulations were conducted using various combinations of wind and ship speeds to examine the precise effects of ship-generated waves on estimating the drift position of a person. Tables 2 and 3 present the drift positions estimated by the two different models, the conventional leeway model and the proposed model for small and large ships under different ship speeds and wind conditions. For combinations of wind speeds of 2, 3, and 4 m/s with ship speeds of 5 kn (2.572 m/s) and 10 kn (5.144 m/s), identical simulation conditions were applied to both small and large ships. After simulating 100 independent drifters, the trajectory coordinates were averaged to obtain the results. Considering the urgency of the rescue operations, drift positions 10 min after release were used as comparison metrics.

Table 2

Results of 10-min drift position estimation under different ship speeds and wind conditions for a small ship (Δ: drift distance, Distance: total diplacement distance=(ΔX)2+(ΔY)2)

Table 3

Results of 10-min drift position estimation under different ship speeds and wind conditions for a large ship (Δ: drift distance, Distance: total diplacement distance=(ΔX)2+(ΔY)2)

An analysis of the results in Tables 2 and 3 reveals that the Kelvin ship-wave–coupled model exhibits two behaviors compared to the conventional Leeway model.

First, when the ship size was fixed and the speed increased from 5 kn (2.572 m/s) to 10 kn (5.144 m/s), the divergent wave component of 82 Yang Zhao, Jung-Ho Kang and Kyung-Chang Lee the wake became more pronounced, and its region of influence expanded. This enhanced wave field induced a stronger lateral flow, leading to a systematic increase in the displacement along the Y-axis. Simultaneously, the wake introduced greater interference with and partial cancellation of the along-track (X-axis) motion, resulting in a reduced X displacement and a smaller total drift distance. This trend was characterized by an increase in Y, a decrease in X, and a reduction in the total displacement distance, defined as the straight-line distance between the initial and final positions. This effect was more pronounced for a large ship whose wake exhibited a higher amplitude and broader spatial footprint, thereby amplifying both the lateral and longitudinal damping effects.

Second, size comparisons at a fixed speed revealed a clear dependence on the speed regime. At 5 kn, the large ship generated a relatively weak lateral induction but maintained a stronger along-track motion, producing a larger X displacement and only minor changes in Y; as a result, the overall drift distance differed little from that predicted by the Leeway model. However, at 10 kn (5.144 m/s), the large ship produced a stronger divergent wave system and imposed more substantial interference on the along-track component, leading to a marked reduction in X displacement and an increase in Y. Consequently, the total drift distance decreased more noticeably, indicating that under high-speed navigation, the disturbance strength and effective range of Kelvin ship-generated waves from large ships were significantly greater than those from small ships.

From the simulation results described above, integrating the wave-induced velocity into the Leeway model consistently altered the short-term drift behavior. For both small and large ships, the lateral (Y-axis) displacement increased, whereas the longitudinal (X-axis) displacement decreased by several tens of meters, depending on the conditions. These effects became more pronounced as ship size and speed increased. This demonstrates that incorporating ship-generated waves in an integrated manner can reduce the initial drifter position errors within the first few minutes after an accident and enable a more accurate determination of the starting point for the search operations.

5. Conclusion and Future Work

In this study, an improved estimation model was proposed to enhance the accuracy of the initial drift position estimation in man-overboard incidents by integrating the wave-induced velocities generated by ships into the Leeway model. For this purpose, the wave height distribution of the ship-generated waves was modeled using the Kelvin model, and the wave-induced velocities calculated from this distribution were incorporated into the Leeway model.

The numerical simulation results showed that the drift positions estimated by the proposed model exhibited a marked increase in lateral (Y-axis) displacement and a tendency for the longitudinal (X-axis) displacement to decrease depending on the conditions. These effects became more pronounced as the ship size and speed increased. These findings demonstrate that incorporating the effects of ship-generated waves in an integrated manner can improve the accuracy of short-term initial drift position estimation.

However, this study did not incorporate human body representation into the simulations. Therefore, the absolute error between the actual and model-estimated drift positions could not be determined. Accordingly, future research will involve verifying the real sea accuracy of the proposed model using GPS-equipped mannequins deployed from operating vessels and evaluating its applicability under various maritime environmental conditions.

Article information Continued