PERFORMANCE COMPARISON OF TOA AND TDOA BASED TRACKING IN UNDERWATER MULTIPATH ENVIRONMENTS USING BERNOULLI FILTER

Underwater localization and tracking is a challenging problem and Time-of-Arrival and Time-Difference-of-Arrival approaches are commonly used. However, the performance difference between these approaches is not well understood or analysed adequately. There are some analytical studies for terrestrial applications with the assumption that the signal arrival times are not correlated. However, this assumption is not valid for underwater propagation. To present the distinct nature of the problem under the water, a high-fidelity simulation is required. In this study, Time-of-Arrival and Time-Difference-of-Arrival approaches are compared using a ray tracing based propagation model. Moreover, basic methods to mitigate the multipath propagation problem are also implemented for Bernoulli filters. Since the Bernoulli filter is a joint detection and tracking filter, the detection performance is also analysed. Comparisons are done for all combinations of filter and measurement approaches. The results can help to design underwater localization systems better suited to the needs.


INTRODUCTION
Unmanned systems are an active area of research, a major branch of which is unmanned underwater vehicles (UUVs) [1].Localization of the UUVs using underwater sensor networks (UWSNs) is one of the most important features of unmanned systems.Due to the dissimilar nature of the underwater environment from the terrestrial, the localization method must also be different.In terrestrial applications, electromagnetic waves are the commonly used form of energy in localization systems.However, electromagnetic waves attenuate very fast under the water.In conductive media, the propagation of electromagnetic signals is only feasible at very low frequencies.Therefore, GPS and similar systems do not function.Acoustic signals can propagate more efficiently under the water and are used commonly for underwater communications and sensor applications [2].An alternative to acoustic sensors for underwater localization may be to use odometry, but due to the integration operation used in filters the sensor noises accumulate in time and the error must be corrected by some other means [3].Hence, UWSNs are the primary choice for the localization of underwater vehicles.The applications of UWSNs are not limited to UUV localization, but also include natural disaster alerts, aided navigation, collection of oceanographic information, surveillance, environmental monitoring, marine and military applications [4].
The localization under the water is susceptible to the estimated speed of sound.The speed of sound under the water is not constant and is affected by environmental variables, such POLISH MARITIME RESEARCH, No 1/2023 136 as temperature, salinity, and pressure.These variables stratify at different depths.Thus, the speed of sound is different at different depth layers, forming a sound speed profile (SSP) [5], [6].Besides the SSP, the propagation direction and attenuation are affected also by the sea-state and bathymetry.Due to these parameters, the acoustic energy tends to concentrate along certain paths.This causes the temporal spread of the pulses, leading to multipath propagation.
Several approaches exist for underwater localization, such as hydrodynamic pressure measurements [7], Angleof-Arrival (AoA), Time-of-Arrival (ToA), Time-Differenceof-Arrival (TDoA), and Received-Signal-Strength (RSS) [8]- [11].The performances of these approaches depend on the applications, environment and sensor types [12].ToA and TDoA are the most commonly used due to their simplicity and low cost [13], [14].ToA and TDoA are also more suitable for omni-directional hydrophone beam patterns and do not require array signal processing.Using hybrid approaches, the information from both approaches can also be fused to reduce the estimation errors [15], [16].
Since ToA and TDoA use similar sensors and hardware, it is important to know which one outperforms the other in certain scenarios.Thus, they are compared and investigated in several works.Under the assumption of zero mean Gaussian measurement noise, it is proved that ToA and TDoA estimates have similar error characteristics [17].However, underwater propagation is more complex than the terrestrial propagation problem [18].In some studies, it is stated that it is not possible to use the algorithms utilized in terrestrial localization without any corrections or modifications for underwater localization [19], [20]reliable simulation \nmethods are required.Ray tracing is one of the alternatives to use \nfor underwater high frequency applications.In this work, we have \nsimulated the effect of different sound speed profiles (SSPs.Despite these remarks, to the best of the author's knowledge, there are no realistic comparisons between ToA and TDoA for underwater localization.The primary goal of this work is to analyse the performances of the ToA and TDoA approaches using realistic simulations.The simulation tool is also important for comparisons.Otherwise, the assumptions for analysis may become too simple and cannot provide the required results with high fidelity. The complexity of the underwater environment and smaller targets make the detection of underwater signals a challenging task.Detection of weak targets in low SNR scenarios has drawn attention in recent years [21], [22]in order to identify the enemy emission sonar source accurately.Using the digital watermarking technology and combining with the good time-frequency characteristics of fractional Fourier transform (FRFT.Unlike radar applications, detection of the target is not the only major problem in underwater tracking.Another important aspect of the underwater localization is the multipath effect.Due to multipath effects, multiple detections can occur from a single source.This is different from the Line-of-Sight (LoS) and Non-LoS (NLoS) ranging encountered in terrestrial applications.Unless the SSP is isovelocity, there is no true LoS path, and all paths are curved in underwater environments.If the distance between the transmitter and receiver nodes is great, the signals also probably bounce from the bottom or surface.This raises the question of which detections to select as measurements.One approach is to exclude outlier detections.The detections are distributed temporally.High deviations in detection times can increase localization error.Hence, the exclusion of outlier detections is proposed by limiting the arrival times using the Random Sample Consensus (RANSAC) algorithm [23].The RANSAC algorithm smoothens data by erasing outliers from the processed data [24]Random Sample Consensus (RANSAC.Maximum likelihood based optimization of location estimation is an alternative approach [25]the time difference of arrival (TDOA.Stochastic optimization algorithms such as the firefly algorithm are also applied to TDoA localization [26]constrained weighted least squares (CWLS.Fusing TDoA and Frequency-Difference-of-Arrival (FDoA) measurements to jointly estimate the target location and the sound speed is also proposed.In this approach, weighted least squares (WLS) is chosen as the optimization approach [27], [28].Changing the network topology and adding mobile sensor nodes to the UWSN are also proposed [29].
Besides multipath, the lack of information on the true sound speed, c, is also an important source of error.In most underwater localization approaches the sound speed, c, is assumed to be known a priori [30], [31].This assumption usually does not hold.If the assumed or estimated sound speed, ĉ, is constant but different from the actual speed, c, then the error increases with increasing range [20].A correction term can be added to the tracking filter to mitigate the mismatch between c and ĉ [32].Using the information about the SSP can also help to improve the estimates [33].But this approach requires complete information of the bathymetry and the existence of a direct path.It uses stochastic optimization to estimate the SSP.The existence of a direct path is not a realistic assumption if the transmitter and the receiver nodes are located at distances that are multiple times the depth.
In this work, no assumptions for bathymetry or SSP are made in the measurement equations.The simulation scenarios are configured using a sensor network composed of a transmitter and four receivers similar to [34].As mentioned above, realistic simulations of pulse arrival times and amplitudes are critical for analysis.Based on the frequency of the signal, the numerical underwater sound propagation solvers are divided into two main categories: low and high frequency solvers [35].There are no hard definitions about how to categorize the problem as high or low frequency.But the rule-of-thumb is that if the depth of the environment is at least several times the wavelength of the acoustic signals the simulation is high frequency.We used ray tracing and the BELLHOP package to calculate the signal arrival times and amplitudes [36], [37].Ray tracing is applied in underwater propagation problems, for which it is well known and recognized [38], [39].
The outputs of the solver are then used as detections in Bernoulli filters.The Bernoulli filter is implemented using Gaussian mixtures (GMs).The Bernoulli filter can jointly detect and track targets [40].It is based on a random finite set (RFS) framework [41].Compared to other target tracking filters, RFS based filters have several advantages such as the inherent ability to detect, track and classify the targets, and not requiring a data association step, which is computationally expensive.Tracking in multipath environments is similar to the extended target tracking (ETT) problem, where multiple detections stem from a single target [42].Unlike the ETT, in multipath problems the targets are assumed as point targets, such as a pinger.In this study we also analyse how to optimally select or fuse the detections.We applied three well-known tracking approaches to the tracking algorithms.The update steps of the Bernoulli filter are modified to resemble the Nearest-Neighbourhood (NN), Probabilistic Data Association (PDA) and the detectoroutput based filters, which are named NN-Bernoulli, PDA-Bernoulli and Bernoulli filters, respectively.To compare the filter performances, the prediction and update steps of the probability of target existence of the filters are based on the Bernoulli filter.Different approaches for track initialization and terminations also exist.
The NN-Bernoulli filter selects a single detection per sensor using maximum likelihood (ML) and filters out the other detections from other paths [43].The PDA-Bernoulli filter updates using all of the measurements and applies moment matching to obtain a single Gaussian distribution [44].Each filter type also has two variants for the ToA and TDoA measurement models.The simulations are run in a flat bathymetry with 40 m depth, like the south of the Sea of Marmara.The optimal topology of the underwater sensor network is another research problem [45].We preferred to exclude this problem and configured a simple geometry but added a scenario with randomized sensor locations for comparison.The scenario results give insight about the impact of the topology of the sensor network on underwater target tracking.The scenarios are based on the localization of an underwater vehicle with a pinger which typically transmits pulses at 25 kHz.Hence, the problem can be classified as high frequency.After obtaining the arrivals, random clutter is also added to the measurement set.The existence of the target and the target state vector are estimated using the filters.The novelties of this work are as follows: 1) This is the first realistic comparison of ToA and TDoA approaches for underwater localization.We show which applications ToA and TDoA are more suitable for.
2) The basic tracking approaches, namely NN, PDA and multiple hypotheses, are applied to decide on how to best select or process the detections.3) Missed detection and clutter detection terms are merged to obtain a filtering formulation applicable to multiple sensors.
The rest of the paper is as follows.In Section 2, the simulation approach and filter formulations are explained.In Section 3, the simulation results are presented and the results are explained.The paper is concluded in Section 4 with some insights and planned work.

PROPAGATION AND MEASUREMENT MODELS RAy TRACING
The underwater propagation is modelled using the BELLHOP package.The scenario environment is set up and the time of pulse arrivals between the pinger and sensor nodes is calculated.A sample of the environment and simulation at 300 m distance is seen in Fig. 1.The SSP is shown in Fig. 2. Munk SSP models the deep ocean and we take 50 m below the surface [46].The detected pulses at a sensor node are given in Fig. 3.

MEASUREMENT MODELS
The output of the ray tracing with the arrival times and amplitudes of the pulses are shown in Fig. 3.The choice of detections is important for filter updates.Since the speed of light is almost constant in air, in ultra-wideband (UWB) applications the earliest detections are assumed to have travelled along the shortest path.Therefore, they are assumed as LoS detections.If a similar approach were adopted, a natural choice would be to pick the first detection at each sensor node [20].If the configuration in Fig. 3 is simulated, the arrival times and the amplitudes in the configured environment are obtained as in Fig. 3.The earliest arrival corresponds to the direct path, which travels without any bounces.When the direct path detection is used and c is assumed as 1500 m/s, the distance found is 290.81 m.When the arrival times are examined, it is seen that a better estimate is the detection that occurred from the bottom bounce.This path yields 301 m.Even if the earliest arrival is a direct path, the speed mismatch causes a relatively large error in distance estimation.As in the example, this problem can be mitigated by using the second detection.In real applications, it is impossible to decide which arrival has the lowest error without complete knowledge of the environmental parameters and it is difficult and expensive to measure all the environmental parameters accurately.Hence, a more robust yet still simple method is proposed to exploit the measurements for practical purposes.
Since in the real world the environmental variables are not known a-priori, no structures can be assumed about the arrival times of the detections.Hence, statistical assumptions as in ETT applications do not apply to this problem [42].A more general approach is to assume that the multipath effects are clutter.It has been shown that the NN, PDA and MHT filters have similar structures to variants of Bernoulli with different approximations [47].In this work, we keep the fundamental ideas of these filters and reformulate the update step of the Bernoulli filter.In the update step, the NN-Bernoulli filter picks only the measurement combination with the ML.The PDA-Bernoulli filter updates with each measurement and uses moment matching to obtain posterior distribution with a single GM component.The Bernoulli filter uses pruning and merging to limit the growing number of components.It makes no other approximations and uses all detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is given as a union of multiple sets: multipath, clutter and Ø, which represents no detections.The unification of all these sets yields the measurement set and is defined as multipath effects are clutter.It has been shown that the NN, PDA and M structures to variants of Bernoulli with different approximations [47].In fundamental ideas of these filters and reformulate the update step of the update step, the NN-Bernoulli filter picks only the measurement combina PDA-Bernoulli filter updates with each measurement and uses mome posterior distribution with a single GM component.The Bernoulli filter use to limit the growing number of components.It makes no other approx detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is given sets: multipath, clutter and ∅, which represents no detections.The unifi yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time step  singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and it hold The Bernoulli filter is based on random finite sets (RFSs).Hence, and measurements are RFSs.The measurement set, Z k , is generated us element combinations of Υ k i .The further processing of inputs differs for To

ToA Measurement Model:
ToA measurement is generated using singleton subset combinatio {∅} is also a singleton subset and indicates no detection.Following the not written as Hence, all hypotheses from each sensor including no detections are combi

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are again but then further subtracted from each other.If a set returns {∅}, the se measurements at that time step.The   set can be written as where multipath effects are clutter.It has been shown that the NN, PDA and structures to variants of Bernoulli with different approximations [47].I fundamental ideas of these filters and reformulate the update step of t update step, the NN-Bernoulli filter picks only the measurement comb PDA-Bernoulli filter updates with each measurement and uses mom posterior distribution with a single GM component.The Bernoulli filter to limit the growing number of components.It makes no other app detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is giv sets: multipath, clutter and ∅, which represents no detections.The un yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time step singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and it ho The Bernoulli filter is based on random finite sets (RFSs).Henc and measurements are RFSs.The measurement set, Z k , is generated element combinations of Υ k i .The further processing of inputs differs for

ToA Measurement Model:
ToA measurement is generated using singleton subset combina {∅} is also a singleton subset and indicates no detection.Following the n written as Hence, all hypotheses from each sensor including no detections are com

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are agai but then further subtracted from each other.If a set returns {∅}, the measurements at that time step.The   set can be written as is the number of arrivals at the ith sensor node at time step k and multipath effects are clutter.It has been shown that the NN, PDA and MHT filters have similar structures to variants of Bernoulli with different approximations [47].In this work, we keep the fundamental ideas of these filters and reformulate the update step of the Bernoulli filter.In the update step, the NN-Bernoulli filter picks only the measurement combination with the ML.The PDA-Bernoulli filter updates with each measurement and uses moment matching to obtain posterior distribution with a single GM component.The Bernoulli filter uses pruning and merging to limit the growing number of components.It makes no other approximations and uses all detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is given as a union of multiple sets: multipath, clutter and ∅, which represents no detections.The unification of all these sets yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time step  and    is also a set of singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and it holds |e| ≤ 1.The Bernoulli filter is based on random finite sets (RFSs).Hence, the target state vectors and measurements are RFSs.The measurement set, Z k , is generated using all possible single element combinations of Υ k i .The further processing of inputs differs for ToA and TDoA.

ToA Measurement Model:
ToA measurement is generated using singleton subset combinations of all Υ   .Note that {∅} is also a singleton subset and indicates no detection.Following the notation in [48],   can be written as Hence, all hypotheses from each sensor including no detections are combined to form   .

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are again generated using Eq. ( 2) but then further subtracted from each other.If a set returns {∅}, the sensor does not generate measurements at that time step.The   set can be written as is also a set of singletons of clutter detections.Note that ∀e∈ multipath effects are clutter.It has been shown that the NN, PDA structures to variants of Bernoulli with different approximations fundamental ideas of these filters and reformulate the update ste update step, the NN-Bernoulli filter picks only the measurement PDA-Bernoulli filter updates with each measurement and uses posterior distribution with a single GM component.The Bernoulli to limit the growing number of components.It makes no othe detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step sets: multipath, clutter and ∅, which represents no detections.T yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set an The Bernoulli filter is based on random finite sets (RFSs).and measurements are RFSs.The measurement set, Z k , is gene element combinations of Υ k i .The further processing of inputs diffe

ToA Measurement Model:
ToA measurement is generated using singleton subset co {∅} is also a singleton subset and indicates no detection.Following written as Hence, all hypotheses from each sensor including no detections ar

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations ar but then further subtracted from each other.If a set returns {∅} measurements at that time step.The   set can be written as is itself a set and it holds |e|≤1.The Bernoulli filter is based on random finite sets (RFSs).Hence, the target state vectors and measurements are RFSs.The measurement set, Z k , is generated using all possible single element combinations of multipath effects are clutter.It has been shown that the NN, PDA structures to variants of Bernoulli with different approximations [4 fundamental ideas of these filters and reformulate the update step update step, the NN-Bernoulli filter picks only the measurement c PDA-Bernoulli filter updates with each measurement and uses posterior distribution with a single GM component.The Bernoulli fi to limit the growing number of components.It makes no other detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step i sets: multipath, clutter and ∅, which represents no detections.Th yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and The Bernoulli filter is based on random finite sets (RFSs).H and measurements are RFSs.The measurement set, Z k , is genera element combinations of Υ k i .The further processing of inputs differ

ToA Measurement Model:
ToA measurement is generated using singleton subset com {∅} is also a singleton subset and indicates no detection.Following written as Hence, all hypotheses from each sensor including no detections are

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are but then further subtracted from each other.If a set returns {∅}, measurements at that time step.The   set can be written as . The further processing of inputs differs for ToA and TDoA.

ToA Measurement Model:
ToA measurement is generated using singleton subset combinations of all multipath effects are clutter.It has been shown that the NN, PDA and structures to variants of Bernoulli with different approximations [47].fundamental ideas of these filters and reformulate the update step of update step, the NN-Bernoulli filter picks only the measurement com PDA-Bernoulli filter updates with each measurement and uses mo posterior distribution with a single GM component.The Bernoulli filte to limit the growing number of components.It makes no other ap detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is g sets: multipath, clutter and ∅, which represents no detections.The u yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time step singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and it h The Bernoulli filter is based on random finite sets (RFSs).Hen and measurements are RFSs.The measurement set, Z k , is generated element combinations of Υ k i .The further processing of inputs differs fo

ToA Measurement Model:
ToA measurement is generated using singleton subset combin {∅} is also a singleton subset and indicates no detection.Following the written as Hence, all hypotheses from each sensor including no detections are com

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are ag but then further subtracted from each other.If a set returns {∅}, the measurements at that time step.The   set can be written as . Note that {Ø} is also a singleton subset and indicates no detection.Following the notation in [48], Z k can be written as Hence, all hypotheses from each sensor including no detections are combined to form Z k .

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are again generated using Eq. ( 2) but then further subtracted from each other.If a set returns {Ø}, the sensor does not generate measurements at that time step.The Z k set can be written as multipath effects are clutter.It has been shown that the NN, PDA and MHT filt structures to variants of Bernoulli with different approximations [47].In this wo fundamental ideas of these filters and reformulate the update step of the Berno update step, the NN-Bernoulli filter picks only the measurement combination w PDA-Bernoulli filter updates with each measurement and uses moment mat posterior distribution with a single GM component.The Bernoulli filter uses prun to limit the growing number of components.It makes no other approximatio detections and returns a mixture of Gaussians.
For the th sensor node, a detection set at the th time step is given as a u sets: multipath, clutter and ∅, which represents no detections.The unification yields the measurement set and is defined as where    is the number of arrivals at the th sensor node at time step  and    singletons of clutter detections.Note that ∀e ∈ Υ k i is itself a set and it holds |e| ≤ The Bernoulli filter is based on random finite sets (RFSs).Hence, the tar and measurements are RFSs.The measurement set, Z k , is generated using all element combinations of Υ k i .The further processing of inputs differs for ToA and

ToA Measurement Model:
ToA measurement is generated using singleton subset combinations of a {∅} is also a singleton subset and indicates no detection.Following the notation in written as Hence, all hypotheses from each sensor including no detections are combined to

TDoA Measurement Model:
TDoA is slightly different from ToA.The combinations are again generat but then further subtracted from each other.If a set returns {∅}, the sensor do measurements at that time step.The   set can be written as For filtering, a motion and measurement model is required.In this work, the target is assumed to move according to the constant-velocity (CV) motion model.The target state vector is then LLI FILTER r filtering, a motion and measurement model is required.In this work, the target is to move according to the constant-velocity (CV) motion model.The target state vector is e coordinates of the target along the  and -axes, respectively.Similarly,   and   are onents of the velocity vector along the  and -axes.The CV model is linear and , so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.re defined as update period of the filter.Each measurement vector for ToA and TDoA is nd  are the sensor number labels and  is the total number of sensors.Note that Eq. (8) 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the with that specific sensor label are deleted.e predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq. ( 9) x, y are the coordinates of the target along the x and y-axes, respectively.Similarly, v x and v y are the components of the velocity vector along the x and y-axes.The CV model is linear and Gaussian, so it can be written as

LLI FILTER
r filtering, a motion and measurement model is required.In this work, the target is to move according to the constant-velocity (CV) motion model.The target state vector is e coordinates of the target along the  and -axes, respectively.Similarly,   and   are onents of the velocity vector along the  and -axes.The CV model is linear and , so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.re defined as update period of the filter.Each measurement vector for ToA and TDoA is nd  are the sensor number labels and  is the total number of sensors.Note that Eq. (8) 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the with that specific sensor label are deleted.he predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq. ( 9) where q k ~N(0,Q), Q is the process error covariance matrix and F is the state transition matrix.F and Q are defined as

LLI FILTER
r filtering, a motion and measurement model is required.In this work, the target is o move according to the constant-velocity (CV) motion model.The target state vector is e coordinates of the target along the  and -axes, respectively.Similarly,   and   are onents of the velocity vector along the  and -axes.The CV model is linear and so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.re defined as pdate period of the filter.Each measurement vector for ToA and TDoA is nd  are the sensor number labels and  is the total number of sensors.Note that Eq. ( 8) 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the with that specific sensor label are deleted.e predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq. ( 9) LLI FILTER r filtering, a motion and measurement model is required.In this work, the target is to move according to the constant-velocity (CV) motion model.The target state vector is e coordinates of the target along the  and -axes, respectively.Similarly,   and   are onents of the velocity vector along the  and -axes.The CV model is linear and so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.re defined as pdate period of the filter.Each measurement vector for ToA and TDoA is nd  are the sensor number labels and  is the total number of sensors.Note that Eq. ( 8) 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the with that specific sensor label are deleted.e predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq.(9) Δt is the update period of the filter.Each measurement vector for ToA and TDoA is

OULLI FILTER
For filtering, a motion and measurement model is required.In this work, the target is d to move according to the constant-velocity (CV) motion model.The target state vector is the coordinates of the target along the  and -axes, respectively.Similarly,   and   are ponents of the velocity vector along the  and -axes.The CV model is linear and n, so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.are defined as e update period of the filter.Each measurement vector for ToA and TDoA is and  are the sensor number labels and  is the total number of sensors.Note that Eq. ( 8) .( 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the ts with that specific sensor label are deleted.The predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq. ( 9)

LLI FILTER
r filtering, a motion and measurement model is required.In this work, the target is to move according to the constant-velocity (CV) motion model.The target state vector is e coordinates of the target along the  and -axes, respectively.Similarly,   and   are onents of the velocity vector along the  and -axes.The CV model is linear and , so it can be written as ∼ (0, ),  is the process error covariance matrix and  is the state transition matrix.re defined as update period of the filter.Each measurement vector for ToA and TDoA is nd  are the sensor number labels and  is the total number of sensors.Note that Eq. ( 8) 9) show the full extent of the vectors.If a sensor returns no detections, i.e., ∅, then the with that specific sensor label are deleted.e predicted measurements are calculated by replacing the  terms in Eq. ( 8) and Eq.(9) where i and m are the sensor number labels and S is the total number of sensors.Note that Eq. ( 8) and Eq. ( 9) show the full extent of the vectors.If a sensor returns no detections, i.e., Ø, then the elements with that specific sensor label are deleted.
The predicted measurements are calculated by replacing the r terms in Eq. ( 8) and Eq. ( 9) with and    are the th sensor node positions along the  and -axes, respectively.tarting with a prior probability density function (PDF)  −1|−1 () at time step , the recursions are performed using the equations below: where h sensor node positions along the  and -axes, respectively.or probability density function (PDF)  −1|−1 () at time step , the using the equations below: and or node positions along the  and -axes, respectively.bability density function (PDF)  −1|−1 () at time step , the ed using the equations below: are the th sensor node positions along the x and y-axes, respectively.
Starting with a prior probability density function (PDF) p k−1|k−1 (x) at time step k, the Bayesian recursions are performed using the equations below: where    and    are the th sensor node positions along the  and -axes, respectively.Starting with a prior probability density function (PDF)  −1|−1 () at time step , the Bayesian recursions are performed using the equations below:

Bernoulli RFS
Using the Finite Set Statistics (FISST) based on the RFSs, Mahler extended the single target Bayesian formulations of Eq. ( 11) and Eq. ( 12) to multi-object tracking equations [41].FISST provides a framework to unify tracking, detection, data association and classification problems [41], [49].
In RFS based approaches, the target states and measurements are handled as sets: 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() = {  (1) ,   (2) , ⋯ , where   and   are the number of targets and the measurements at time , ℱ() and ℱ() are the set of all finite subsets of state and measurement spaces  and , respectively.The primary objective of this paper is to compare the ToA and TDoA approaches for underwater detection and tracking.A single target tracking simulation is sufficient to show the performances.The Bernoulli filter is an optimal filter derived from the FISST framework and suitable for on/off switching of dynamic systems [40], [41].It models the target state as a Bernoulli RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = (|| = ) and a spatial distribution for set elements (11) where    and    are the th sensor node positions along the  and -axes, respectively.Starting with a prior probability density function (PDF)  −1|−1 () at time step  Bayesian recursions are performed using the equations below:  12).

Bernoulli RFS
Using the Finite Set Statistics (FISST) based on the RFSs, Mahler extended the si target Bayesian formulations of Eq. ( 11) and Eq. ( 12) to multi-object tracking equations FISST provides a framework to unify tracking, detection, data association and classific problems [41], [49].
In RFS based approaches, the target states and measurements are handled as sets: 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() = {  (1) ,   (2) , ⋯ , where   and   are the number of targets and the measurements at time , ℱ() and ℱ( the set of all finite subsets of state and measurement spaces  and , respectively.The primary objective of this paper is to compare the ToA and TDoA approache underwater detection and tracking.A single target tracking simulation is sufficient to show performances.The Bernoulli filter is an optimal filter derived from the FISST framework suitable for on/off switching of dynamic systems [40], [41].It models the target state as a Bern RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = (|| = ) a spatial distribution for set elements where π k|k−1 (x k |x k−1 ) is the Markov transition density, p k|k (x k | z 1:k ) is the posterior, and g k (z k | x k ) the likelihood density.
The expected a posteriori (EAP) or the maximum a posteriori (MAP) estimates of the target state can be calculated using p k|k (x k |z 1:k ), which is the result of Eq. ( 12).

BERNOULLI RFS
Using the Finite Set Statistics (FISST) based on the RFSs, Mahler extended the single target Bayesian formulations of Eq. and Eq. to multi-object tracking equations [41].FISST provides a framework to unify tracking, detection, data association and classification problems [41], [49].
In RFS based approaches, the target states and measurements are handled as sets: where    and    are the th sensor node positions along the  and -axes, resp Starting with a prior probability density function (PDF)  −1|−1 () Bayesian recursions are performed using the equations below:  12).
In RFS based approaches, the target states and measurements are handl 1) ,   (2) , ⋯ ,   (  ) ∈ ℱ()   = {  (1) ,   (2) , ⋯ , where   and   are the number of targets and the measurements at time , ℱ the set of all finite subsets of state and measurement spaces  and , respectiv The primary objective of this paper is to compare the ToA and TD underwater detection and tracking.A single target tracking simulation is suf performances.The Bernoulli filter is an optimal filter derived from the FIS suitable for on/off switching of dynamic systems [40], [41].It models the target RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = spatial distribution for set elements where    and    are the th sensor node positions along the  and -axes, resp Starting with a prior probability density function (PDF)  −1|−1 () Bayesian recursions are performed using the equations below:  12).
In RFS based approaches, the target states and measurements are handl 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ()   = {  (1) ,   (2) , ⋯ , where   and   are the number of targets and the measurements at time , ℱ the set of all finite subsets of state and measurement spaces  and , respectiv The primary objective of this paper is to compare the ToA and TD underwater detection and tracking.A single target tracking simulation is suf performances.The Bernoulli filter is an optimal filter derived from the FIS suitable for on/off switching of dynamic systems [40], [41].It models the target RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = spatial distribution for set elements where M k and N k are the number of targets and the measurements at time k, where    and    are the th sensor node positions along the  and -axes, respectively.Starting with a prior probability density function (PDF)  −1|−1 () at time step , the Bayesian recursions are performed using the equations below: where π |−1(   | | −1 ) is the Markov transition density,  | (   | |  : ) is the posterior, and   (   | |   ) the likelihood density.The expected a posteriori (EAP) or the maximum a posteriori (MAP) estimates of the target state can be calculated using  | (   | |  1: ), which is the result of Eq. ( 12).

Bernoulli RFS
Using the Finite Set Statistics (FISST) based on the RFSs, Mahler extended the single target Bayesian formulations of Eq. ( 11) and Eq. ( 12) to multi-object tracking equations [41].FISST provides a framework to unify tracking, detection, data association and classification problems [41], [49].
In RFS based approaches, the target states and measurements are handled as sets: 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() = {  (1) ,   (2) , ⋯ , where   and   are the number of targets and the measurements at time , ℱ() and ℱ() are the set of all finite subsets of state and measurement spaces  and , respectively.The primary objective of this paper is to compare the ToA and TDoA approaches for underwater detection and tracking.A single target tracking simulation is sufficient to show the performances.The Bernoulli filter is an optimal filter derived from the FISST framework and suitable for on/off switching of dynamic systems [40], [41].It models the target state as a Bernoulli RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = (|| = ) and a spatial distribution for set elements and where    and    are the th sensor node positions along the  and -axes, respectively.Starting with a prior probability density function (PDF)  −1|−1 () at time st Bayesian recursions are performed using the equations below: is the Markov transition density,  | (   | |  : ) is the poste   (   | |   ) the likelihood density.The expected a posteriori (EAP) or the maximum a p (MAP) estimates of the target state can be calculated using  | (   | |  1: ), which is the Eq. ( 12).

Bernoulli RFS
Using the Finite Set Statistics (FISST) based on the RFSs, Mahler extended t target Bayesian formulations of Eq. ( 11) and Eq. ( 12) to multi-object tracking equati FISST provides a framework to unify tracking, detection, data association and clas problems [41], [49].
In RFS based approaches, the target states and measurements are handled as sets 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ()   = {  (1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() where   and   are the number of targets and the measurements at time , ℱ() and ℱ the set of all finite subsets of state and measurement spaces  and , respectively.The primary objective of this paper is to compare the ToA and TDoA approa underwater detection and tracking.A single target tracking simulation is sufficient to performances.The Bernoulli filter is an optimal filter derived from the FISST framew suitable for on/off switching of dynamic systems [40], [41].It models the target state as a B RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = (|| = spatial distribution for set elements are the set of all finite subsets of state and measurement spaces where    and    are the th sensor node positions along the  and -axes, respe Starting with a prior probability density function (PDF)  −1|−1 () a Bayesian recursions are performed using the equations below:
In RFS based approaches, the target states and measurements are handle 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() = {  (1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() where   and   are the number of targets and the measurements at time , ℱ( the set of all finite subsets of state and measurement spaces  and , respective The primary objective of this paper is to compare the ToA and TDo underwater detection and tracking.A single target tracking simulation is suffi performances.The Bernoulli filter is an optimal filter derived from the FISS suitable for on/off switching of dynamic systems [40], [41].It models the target s RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = spatial distribution for set elements Eq. ( 12).
In RFS based approaches, the target states and measurements are   = {  (1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() = {  (1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() where   and   are the number of targets and the measurements at tim the set of all finite subsets of state and measurement spaces  and , re The primary objective of this paper is to compare the ToA an underwater detection and tracking.A single target tracking simulation performances.The Bernoulli filter is an optimal filter derived from th suitable for on/off switching of dynamic systems [40], [41].It models the RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ spatial distribution for set elements The primary objective of this paper is to compare the ToA and TDoA approaches for underwater detection and tracking.A single target tracking simulation is sufficient to show the performances.The Bernoulli filter is an optimal filter derived from the FISST framework and suitable for on/off switching of dynamic systems [40], [41].It models the target state as a Bernoulli RFS.The PDF of an RFS, X, is defined by a cardinality distribution ρ(n)=P(|X|=n) and a spatial distribution for set elements where    and    are the th sensor node positions along the  and -axes, respe Starting with a prior probability density function (PDF)  −1|−1 () a Bayesian recursions are performed using the equations below:
In RFS based approaches, the target states and measurements are handle 1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ()   = {  (1) ,   (2) , ⋯ ,   (  ) } ∈ ℱ() where   and   are the number of targets and the measurements at time , ℱ( the set of all finite subsets of state and measurement spaces  and , respective The primary objective of this paper is to compare the ToA and TDo underwater detection and tracking.A single target tracking simulation is suffi performances.The Bernoulli filter is an optimal filter derived from the FISS suitable for on/off switching of dynamic systems [40], [41].It models the target s RFS.The PDF of an RFS, , is defined by a cardinality distribution ρ() = spatial distribution for set elements where n is the cardinality and p n (X) is the spatial distribution.When ρ(n) is assumed Bernoulli, then X has either one element or is Ø.Thus, Bernoulli RFS is obtained.It can be written as where  is the cardinality and   () is the joint spatial distribution.When Bernoulli, then  has either one element or is ∅.Thus, Bernoulli RFS is obtaine as where  is the probability of target existence and () is the spatial PDF of the

Prediction and Update Steps
() can be approximated using sequential Monte Carlo (SMC) or G (GMs) [40], [41], [50].In this work, the GM approach is utilized.Then, the pre are approximated as where q is the probability of target existence and p(x) is the spatial PDF of the target.

RESULT
For comparison, three scenarios are created.I where the sensor nodes are positioned at the corners I, the SSP is assumed constant during the scenario and II, the SSP is randomized.Gaussian noise with varian possible  set as 1450 m/s.A different noise is genera The aim of this scenario is to rule out the bias cance For example, if ̂> ̅ then the range estimates have a subtracting range estimates from each other.Scena positions.This configuration negates the bias cancella term in Eq. ( 22) is modified such that where () is the clutter distribution,  is the average number of clutter detections,  Jacobian of the measurement function and  is the measurement error covariance matrix.Eq. ( 20) has two discrete terms.The first term updates the predicted distributi detections occur, the second term updates with detections.In the scenarios, some of the may generate detections and other sensors may not.A new formulation is required to c these two cases.Our formulation of   simultaneously includes detections and missed de at different sensors.The no detection term in Eq. ( 20) is not used and missed detect incorporated into the likelihood function.The g k (i) (z) term in Eq. ( 22) is modified such th ),    |−1 where   is the number of sensors detecting the target.This equation captures all the combinations, since ℎ  returns only the rows corresponding to the active sensors dimensions change accordingly.

RESULTS
For comparison, three scenarios are created.In Scenarios I and II, a test range is where the sensor nodes are positioned at the corners of a rectangle, as seen in Fig. 4. In S I, the SSP is assumed constant during the scenario and between the pinger and sensors.In S II, the SSP is randomized.Gaussian noise with variance 1.0 is added to the SSP with the m possible  set as 1450 m/s.A different noise is generated at each step and from each sens The aim of this scenario is to rule out the bias cancellation that occurs in TDoA measur For example, if ̂> ̅ then the range estimates have a positive bias.TDoA mitigates this subtracting range estimates from each other.Scenario III is configured with random positions.This configuration negates the bias cancellation property of the TDoA approach where n d is the number of sensors detecting the target.This equation captures all the possible combinations, since h k returns only the rows corresponding to the active sensors and the dimensions change accordingly.

RESULTS
For comparison, three scenarios are created.In Scenarios I and II, a test range is created where the sensor nodes are positioned at the corners of a rectangle, as seen in Fig. 4. In Scenario I, the SSP is assumed constant during the scenario and between the pinger and sensors.In Scenario II, the SSP is randomized.Gaussian noise with variance 1.0 is added to the SSP with the minimum possible c set as 1450 m/s.A different noise is generated at each step and from each sensor node.The aim of this scenario is to rule out the bias cancellation that occurs in TDoA measurements.For example, if ĉ>c then the range estimates have a positive bias.TDoA mitigates this bias by subtracting range estimates from each other.Scenario III is configured with random sensor positions.This configuration negates the bias cancellation property of the TDoA approach.The average clutter rate, λ, is taken as 1 in the simulations.Clutter is generated and added independently at each sensor node.Due to multipath arrivals, a higher amount of average clutter is observed at the sensors.Thus, c(z)=1/1000 and λ=5.These values are the same for both ToA and TDoA measurements.But due to the higher number of measurement combinations in TDoA, p b is 0.02 for ToA and 0.001 for TDoA.
The performance of the filters is investigated using an optimal subpattern assignment (OSPA) metric [51].The OSPA distance captures both state vector estimates and target existence probability estimates.The OSPA distance between the estimated target set and the ground truth is given as Not all pulse arrivals are used as detections.The detection threshold is assumed -60 dB and wn in Fig. 3.At each time step, between each pinger and sensor node the propagation ulation is done, and detections are extracted.The target is generated at a random point on the le with a radius of 500 m with the centre at the origin.The direction of the target is pointed ards the centre.The speed of the target is also randomized at the beginning of the scenario as ∼ (5.0,20.0).The target is introduced at the 20th time step and continues for 170 steps.Δ = sec is the update period.
The average clutter rate, λ, is taken as 1 in the simulations.Clutter is generated and added ependently at each sensor node.Due to multipath arrivals, a higher amount of average clutter bserved at the sensors.Thus, () = 1/1000 and λ = 5.These values are the same for both and TDoA measurements.But due to the higher number of measurement combinations in oA,   is 0.02 for ToA and 0.001 for TDoA.
The performance of the filters is investigated using an optimal subpattern assignment PA) metric [51].The OSPA distance captures both state vector estimates and target existence bability estimates.The OSPA distance between the estimated target set and the ground truth is en as where γ is a cutoff parameter, d(x,z)=|x−z| is the distance between single target states, p ≥1 is the order of norm, and d γ (x,z)=min{γ,d(x,z)} sets the cutoff for state estimate errors.γ indicates the estimate and ground truth combination with the lowest distance.
The mean OSPA errors of 300 Monte Carlo runs are shown in Fig. 5 and Fig. 6.In Scenario I, there are almost no false alarms in the first 20 steps.There is a sharp increase at the 21st time step due to the introduction of a target.The target is detected after 2 steps with a sudden decrease in errors.The target moves towards the centre of the sensor range.Filters with TDoA measurements yield lower errors while the target is in the rectangle formed by the sensors.This is due to the bias in range estimates.When the target is between the sensors, the estimation biases are subtracted from each other.Hence, the TDoA measurements have less bias than ToA.Despite this advantage, when the target gets far away, the cancellation cannot negate the biases.It is known that TDoA localizes the targets on hyperbolas, whereas ToA on circles.Hence, TDoA yields higher errors.In Scenario I, there is no clear superior approach among the TDoA measurement based filters.When the target is close to leaving the test range at about the 100th step, the errors increase.When the target is outside the test range, the NN-Bernoulli filter with ToA measurements has higher performance than the other filters.The other ToA measurement based filters also perform better than the TDoA based filters.On the contrary, the NN-Bernoulli filter with TDoA measurements performs the worst.This simulation indicates that when a test range is set up with relatively stable environmental parameters, TDoA measurements outperform the ToA measurement based filters.If the sensor nodes are randomized or the target is not between the sensor nodes, the ToA measurement based filters perform better.NN based measurement selection simplifies the implementation and hastens the computations since only one hypothesis remains after ML based selection of the measurement.
Scenario II is a non-stationary environment due to the randomized SSP.This can also be seen in the ripples in the errors in Fig. 6.Except at the middle of the test range, NN-Bernoulli with ToA measurement outperforms the other filters in all positions.The Bernoulli filter with TDoA measurements and the ToA measurement based filters have a clear advantage over TDoA based filters.

CONCLUSION
In this study, we compared the performances of ToA and TDoA based underwater localization performances.To generalize the problem, we used the Bernoulli filter, which can jointly estimate the existence and the state of the target.Moreover, selection or merging of measurements is also tested by reformulating the Bernoulli filter update step.The filters are tested under two different scenarios and the performances are provided with the OSPA distance.The results indicate that ToA measurements with the NN approach yield better results on average.In a controlled environment, TDoA based filters show similar performance, which is better than the ToA measurements.
We plan to expand the NN, PDA and standard Bernoulli filtering approach using localization approaches used in NLoS multipath scenarios in UWB sensors.Also, multi-static problems will be simulated.Since the ray tracing simulations do not provide raw signals, track-before-detect strategies cannot be applied.But smoothing filters will be applied to improve the tracking performances of the filters.

Fig. 1 .Fig. 2 .
Fig. 1.Ray tracing results when the RX and TX nodes are positioned 300 m away from each other.They are both at 10 m depth.The figure shows the rays arriving at the RX node through multiple paths

Fig. 3 .
Fig. 3. Pulse arrival times.The pulses are shown with blue, the threshold level is shown with the green line.

Fig. 4 .
Fig. 4. The locations of sensors and the target