1. Introduction

Accurately locating underwater acoustic sources is essential for a variety of marine-related applications such as environmental monitoring, marine biology and underwater security operations. These applications range from tracking marine animals to aiding the navigation of autonomous underwater vehicles (AUVs), as demonstrated by Modalavalasa (Reference Modalavalasa2015). Among the several localisation techniques, Time Difference of Arrival (TDOA) is widely adopted due to its simplicity and effectiveness in source localisation (Xu, Reference Xu2012). However, deploying TDOA in underwater environments poses several practical challenges. The underwater medium is highly unpredictable and complex, which makes precise localisation difficult (Jensen, Reference Jensen2011). For instance, environmental noise, signal reflection and multipath propagation severely affect the measurement accuracy of time delays (Shen and Ding, Reference Shen and Ding2014). Multipath effects occur when signals reflect off the sea surface, the seabed or other underwater structures, leading to overlapping signal paths and distorted arrival times (Touze and Tourneret, Reference Le Touze and Tourneret2008). Furthermore, environmental variables such as water temperature, pressure and salinity significantly influence the speed of sound underwater, thereby altering signal propagation characteristics (Jensen, Reference Jensen2011). Misaligned sensors or calibration drift can further degrade performance, especially in long-term deployments (Chaitanya, Reference Chaitanya2016). Hyperbolic multilateration techniques have also been evaluated for their positioning errors in underwater environments (Chaitanya, Reference Chaitanya2016). In addition to environmental factors, synchronisation issues between sensors complicate TDOA-based methods. Xu (Reference Xu2012) proposed a method to address localisation without requiring time synchronisation, highlighting the potential for improved flexibility. Several studies have incorporated Kalman filters to improve TDOA estimation, such as the work by Laveti on GPS position refinement (Laveti, Reference Laveti2016a) and comparative studies of least squares, SVD and QRD methods for positioning (Munoz, Reference Munoz2008). A significant body of research focuses on improving the robustness of localisation systems in the presence of such uncertainties. For example, Shen and Ding (Reference Shen and Ding2014) explored multiple source localisation using Time of Arrival (TOA) measurements, Touze and Tourneret (Reference Le Touze and Tourneret2008) studied localisation with a single hydrophone, highlighting the value of robust modelling in sparse sensor environments. GPS positioning in multipath environments has also been well documented (Soubielle, Reference Soubielle2002), showing parallels to underwater acoustic challenges. Recent approaches focus on combining optimisation-based frameworks with traditional signal processing to mitigate these problems. By integrating Optimal Recursive Estimation and least-squares optimisation, these frameworks aim to reduce error in noisy conditions (Laveti, Reference Laveti2016b; Munoz, Reference Munoz2008). Kalman filtering strategies have proven particularly useful in improving GPS-based localisation performance (Kumar and Dutt, Reference Kumar and Dutt2019), especially in estimating satellite clock and orbital errors (Bidikar, Reference Bidikar2014).To accurately model sound propagation in water, works such as those by Jensen (Reference Jensen2011) and Schmidt and Finn (Reference Schmidt and Finn2012) have made significant contributions through computational ocean acoustics, providing the basis for accurate environmental modelling in localisation. Advanced Kalman filter variants – such as the cubature Kalman filter (CKF) (Liu et al., Reference Liu, Wei-xin and Pin2012) and comparative analyses of EKF versus particle filters (Ko and Kim, Reference Ko and Kim2012) – have further improved underwater tracking performance. Ribeiro (Reference Ribeiro2004) and Elmezain et al. (Reference Elmezain, Saoud, Sultan, Heshmat, Seneviratne and Hussain2025) also provide comprehensive derivations of Kalman filtering principles and their extensions for nonlinear systems. Linearised approaches such as least-squares correction techniques have also shown promise in practical systems. Huang (Reference Huang2001) proposed a linear-correction least-squares approach for passive localisation, while Ampeliotis and Berberidis (Reference Ampeliotis and Berberidis2008) applied energy-based least squares in acoustic environments. The foundational mathematics of nonlinear and optimal least squares have been elaborated by Teunissen (Reference Teunissen1990) and Cohen and Migliorati (Reference Cohen and Migliorati2017). With optimisation playing a growing role in localisation algorithms, modern gradient descent techniques are being explored for their efficiency. Ruder (Reference Ruder2016) provided a foundational overview of gradient descent methods, which was extended in recent works focusing on large step sizes and non-monotonicity benefits (Wu, Reference Wu2024). Complexity analysis of such optimisation problems has been explored since the 1980s, notably by Krentel (Reference Krentel1986). Recent innovations also apply Kalman filtering in various domains, such as human tracking (Jung, Reference Jung2024), aerospace target estimation (Wei, Reference Wei2024), object detection (Zeng et al., Reference Zeng, Li, Zhou and Zhang2025), bee tracking (Kongsilp et al., Reference Kongsilp, Taetragool and Duangphakdee2024) and robotic precision tracking (Liu and Li, Reference Liu and Li2024). Hybrid filtering strategies have emerged, such as radar-based deep learning fused with unscented Kalman filters (Patrick, Reference Patrick2024) and beam tracking using hybrid Kalman approaches (Zhang, Reference Zhang2024). Iterated unscented Kalman filters also offer better passive tracking performance under certain conditions (Zhan and Wan, Reference Zhan and Wan2007). In the field of visual and geometric tracking, Stenger et al. (Reference Stenger, Mendonça and Cipolla2001) applied unscented Kalman filtering for hand modelling, while Rydell (Reference Rydell2025) developed a complexity-reduction framework for triangulation with approximation bounds. Triangulation techniques continue to evolve, not only in engineering but also in multidisciplinary research including tourism and management (Heale and Forbes, Reference Heale and Forbes2013; Cornelissen, Reference Cornelissen2025; Manthiou et al., Reference Manthiou and Kuppelwieser2025). Finally, with the growing importance of underwater communication, recent works by Yue (Reference Yue2025) provide Monte Carlo-based modelling for non-line-of-sight underwater optical communication, further emphasising the need for reliable localisation frameworks in complex underwater environments. In light of these insights, this paper proposes a robust, adaptive TDOA-based localisation framework that integrates filtering, environmental modelling and optimisation to deliver consistent and accurate source localisation in challenging underwater conditions. Future directions may include the integration of machine learning-based estimators or real-time adaptive filters to further enhance the system’s performance in dynamic and uncertain environments.

1. A. Hybrid localisation techniques (TOA + AOA and TDOA + AOA)

In addition to pure TDOA multilateration, hybrid schemes combine time-based measurements with Angle of Arrival (AOA) to improve observability and reduce ambiguity in multipath/noise-limited conditions. Two commonly used hybrids are: (i) TOA + AOA and (ii) TDOA + AOA. These are briefly summarised here to address alternative approaches and enable a fair comparison with the proposed method.

• • TOA + AOA. Each receiver i measures absolute Time-of-Arrival t _i (requires synchronisation) and bearing θ _i (AOA) of the arriving wavefront. TOA constrains the source to a sphere/circle around each receiver, while AOA constrains it to a ray/line. The joint estimate can be obtained by weighted least squares (WLS) over both range and bearing residuals.

• • TDOA + AOA. Receivers measure TDOA values Δt _i1 relative to a reference receiver 1 (no global synchronisation) together with AOA bearings θ _i. The combined cost function minimises (a) TDOA residuals of the form (‖s−r _i‖−‖s−r ₁‖)/c − Δt _i1 and (b) AOA residuals of the form atan²(y−y _i, x−x _i) − θ _i.

Practical note: hybrid methods typically require directional hydrophones or array processing to estimate AOA, which increases hardware cost and calibration effort. In contrast, the proposed framework improves robustness using signal processing and optimisation on conventional TDOA measurements.

1. B. Optimal recursive estimation

The Kalman filter is a mathematical algorithm used for optimal recursive estimation. It efficiently estimates the state of a dynamic system from noisy observations over time (Liu et al., Reference Liu, Wei-xin and Pin2012). This method is particularly useful to estimate the state of a system that is constantly changing and is influenced by uncertainty. It is widely applied in various fields, including robotics, navigation systems, signal processing and economics. The Kalman filter operates recursively, which means that at each time step, it uses the previous state estimate and the new incoming measurements to update the current estimate (Ko and Kim, Reference Ko and Kim2012). It combines predictions and corrections in a way that minimises the estimation error (Shen and Ding, Reference Shen and Ding2014).

1. Step 1. State vector. The state of a system is typically represented as a vector, denoted by ${X_k}$ , where k is the discrete time step, this state vector contains all the variables that describe the system at a given time.

2. Step 2. Prediction and correction.

   1. 2.1. Prediction. At each step, the Kalman filter predicts the next state based on a model of the system’s dynamics.

      1. State Prediction:

         ${X_k} = {F_k}{X_{k - 1}} + {B_k}{U_k}$

      2. Covariance prediction:

         ${P_k} = {F_k}{P_{k - 1}}F_k^T{Q_k}$
         where ${F_k}$ is the state transition matrix, ${B_k}$ is the control input matrix, ${U_k}$ is the control input, ${P_k}$ is the predicted covariance and ${Q_k}$ is the Covariance.

   2. 2.2. Correction: The prediction is then corrected using new measurements. The correction considers the uncertainty in both the system’s model and the measurement process.

      1. Kalman gain:

         ${K_k} = {P_k}H_k^T{\left( {{H_k}{P_k}{H_k} + {R_k}} \right)^{ - 1}}$

      2. State update:

         ${X_k} = {X_{}} + \left( {{Z_k} - {H_k}{X_k}} \right)$

      3. Covariance update:

         ${P_k} = \left( {I - {K_k}{H_k}} \right){P_k}$
         where ${K_k}$ is the Kalman gain, I is the identity matrix, ${H_k}$ is the measurement matrix and ${K_k}$ is the measurement model.

3. Step 3. Process and measurement models.

   1. 3.1. Process model. This model describes how the system evolves over time. It is typically linear and can be written as

      ${X_k} = {F_k}{X_{k - 1}} + {B_k}{U_k} + {w_k}$
      where ${w_k}$ is the process noise, which is assumed to be Gaussian with zero mean and covariance ${Q_k}$ .

   2. 3.2. Measurement model. This model relates the observed measurements to the true state of the system. It can be expressed as

      ${Z_k} = {H_k}{X_k} + {V_k}$
      where ${V_k}$ is the measurement noise, assumed to be Gaussian with zero mean and covariance ${R_k}.$

4. Step 4. Covariance.

   The Kalman filter maintains a covariance matrix ${P_k}$ which represents the uncertainty in the system’s state estimate. The covariance evolves over time and is used to weigh the trust in the prediction versus the measurement update. The Kalman filter is optimal in the sense that it minimises the mean squared error of the state estimate, assuming the process noise ${w_k}$ and measurement noise ${V_k}$ are Gaussian and have known covariances, the system is linear, and the initial state and covariance are known or can be approximated. In the case where the system or measurements are nonlinear, the Kalman filter can be extended to the extended Kalman filter (EKF) (Huang, Reference Huang2001), which linearises the system at each step.

1. C. Least-squares optimisation

Least-squares optimisation is a powerful mathematical technique used to find the best-fit solution to an overdetermined system, where the number of equations exceeds the number of unknowns. The primary goal of least squares is to minimise the sum of the squared differences between the observed values and the predicted values from a model (Elmezain et al., Reference Elmezain, Saoud, Sultan, Heshmat, Seneviratne and Hussain2025). For a simple linear regression, the objective is to minimise the sum of the squared differences between the observed values ${y_i}$ and the model predictions ${z_i}$ ,

$f\left( x \right) = \sum\limits_{i = 1}^n {{{({y_i} - {z_i})}^2}} $

where n = number of data points.

1. 1. Linear least squares.

   Linear models are commonly used to establish a relationship between observed data and unknown parameters. A linear system can be represented in the following way:

   ${Y=Ax}+\varepsilon$
   where Y is a vector of observed values, A is a matrix representing the design matrix, x is the vector of unknown parameters and $\;\varepsilon \;$ is the error vector.

   The least squares problem then becomes

   $mi{n_x}\left| {\left| {Ax - Y} \right|} \right|_2^2$
   where ${\left| {\left| . \right|} \right|_2}$ represents the Euclidean norm. The solution to the linear least-squares problem can be found using the normal equation
   ${X}={({A^T}A)^{ - 1}}{A^T}Y$

   where X is the optimal parameter that minimises the squared error.

2. 2. Nonlinear least squares.

   In many real-world scenarios, relationships between variables can be nonlinear. In such cases, the least-squares method can still be applied, but it becomes necessary to address a nonlinear optimisation problem (Ampeliotis and Berberidis, Reference Ampeliotis and Berberidis2008). The objective remains to minimise the sum of squared residuals, although the model now includes nonlinear functions of the parameters. The general form of the nonlinear least-squares problem is

   $mi{n_\theta }\mathop \sum \limits_{i = 1}^n {({y_i} - f\left( {{x_i},\theta } \right))^2}$
   where f( ${x_i},\theta )$ is the nonlinear function that predicts the value ${z_i}$ given the parameters θ. With advanced methods for solving both linear and nonlinear problems, least-squares optimisation remains a core technique in applied mathematics and many engineering disciplines (Teunissen, Reference Teunissen1990).