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In this paper, we analyze the problem of acoustic ranging between sensor nodes in an underwater environment. The underwater medium is assumed to be composed of multiple isogradient sound speed profile (SSP) layers where in each layer the sound speed is linearly related to the depth. Furthermore, each sensor node is able to measure its depth and can exchange this information with other nodes. Under these assumptions, we first show how the problem of underwater localization can be converted to the traditional rangebased terrestrial localization problem when the depth information of the nodes is known a priori. Second, we relate the pairwise time of flight (ToF) measurements between the nodes to their positions. Next, based on this relation, we propose a novel ranging algorithm for an underwater medium. The proposed ranging algorithm considers reflections from the seabed and sea surface. We will show that even without any reflections, the transmitted signal may travel through more than one path between two given nodes. The proposed algorithm analyzes them and selects the fastest one (first arrival path) based on the measured ToF and the nodes’ depth measurements. Finally, in order to evaluate the performance of the proposed algorithm we run several simulations and compare the results with other existing algorithms.
Available wireless sensor network (WSN) localization techniques rely on mutual distances between sensors [
In [
In our earlier work [
In [
In [
In this work, we analyze the acoustic signal propagation between two sensor nodes in an underwater environment. We use a raytracing approach to model the propagation which is a valid approximation for highfrequency signal transmission [
The rest of the paper is organized as follows. We describe the network model in Section 2, and we compute the ToF
Consider
Assume that the wave speed in a Cartesian coordinate system is a function of the position and is defined by
In an underwater medium where the sound speed varies only with depth, the availability of depth information does not only allow us to work with only three anchors, it also opens the door to convert the timebased localization problem into a traditional rangebased one as explained next.
Since the SSP is only a function of depth, the problem of ray tracing between two nodes has a cylindrical symmetry around the line parallel to the
The conversion of a 3D underwater localization problem to a 2D one can now be explained as depicted in
In order to relate the ToF to the node positions, we first require to find which ray departing the source reaches a specific destination. In this section, we analytically find the rays that can travel between two nodes with known positions, and we compute their corresponding ToFs based on their trajectories. It is worth mentioning that in an underwater medium with fixed SSP, each ray departing a source can be uniquely characterized by its departing angle.
Here the SSP is considered as a piecewise linear function of the depth which is a valid approximation according to the measured data [
In a single layer, each truncated ray (indexed by
The relation between the ToF and the node positions can then be extracted from a set of differential equations characterized by Snell’s law [
The simple linear ToF approximation based on the depth information between two nodes in the same layer, which assumes a straightline ray propagation, can be derived as
Fermat’s principal, which also leads to Snell’s law, states that the path traveled by a ray between two points is the path that can be traversed in the least amount of time. Therefore, the approximated time based on a straightline ray propagation,
We start our analysis by considering a ray traveling between two points in adjacent layers. As illustrated in
For a twopart ray, the combination of
To simplify the multilayer analysis, we define the concept of
The procedure of ToF computation, as a function of the node positions for a ray which has multiple singlelayer parts is the same as the twopart ray, but with more boundary equations. The combination of all these equations may form a higher order polynomial root finding problem, which consequently may increase the number of ways that a ray can travel between the two points. To predict how a ray may travel inside a multilayer underwater area we introduce several lemmas bellow.
In this way, it can be understood that if Δ
Under the assumption of a perfect reflection, the reflected parts in different layers have the same properties as the corresponding nonreflected parts but with an axial symmetry around the line parallel to the
Thanks to the piecewise linear behavior of the SSP, we are now able to predict how a ray, which starts from a given point, can travel through different layers to arrive at a specific point. Having built a
For instance, as illustrated in
A ray may travel directly from point U, crossing the boundary once, and not leaving the two adjacent layers. Of course, as formulated before, even with such a limitation, there are at most three paths that can be included in this category.
A ray may travel directly from point U, crossing the boundary multiple times, not leaving the two adjacent layers. For three crossings of the ray with the boundary, a fourpart ray will be obtained. Based on
It is also possible that a ray from U to V passes other layers. For instance, in the scenario depicted in
Similar to the fourth layer, this phenomenon may also happen in the first layer which forms the fourth category of traveling paths.
In
For the same scenario as described above,
The above analysis indicates that if a UASN is forced to utilize a straightline propagation model due to any reason (e.g., system complexity) to achieve more accurate localization results in a noisefree environment, it is suggested that sensors are deployed in such a way that the angles of the straight lines between the nodes become large. However, when noisy measurements exist, we should also consider their influence on the mapped horizontal distances.
To our best knowledge, the algorithm in [
Assume that, at a specific depth, the ToF of the fastest ray is a monotonic function of the horizontal range. In other words, a propagating wave at a specific depth reaches the destination with a smaller horizontal distance faster. Then, using the ToF and depth measurements, we can find the horizontal distance through a root finding algorithm such as Newton’s method or bisection. Newton’s method is very fast, but it requires the derivative of the ToF w.r.t. the horizontal distance which is hard to compute. The bisection method is robust, and it eventually finds the solution. However, it requires an upper and a lower bound on the horizontal distance. The lower bound can be set to zero, and the upper bound can be computed through multiplying the measured ToF by the maximum sound speed of the entire environment. In spite of the fact that other efficient numerical rootfinding algorithms can also be used, we utilize the simple bisection algorithm for the results in the simulation section.
Proposed Algorithm.
Compute horizontal distance upper and lower bounds, 


Initialize loop parameters, 
e = 

Compute the average value of the upper and the lower bound,

Find the smallest ToF for this horizontal distance
 Form all possible ray patterns hosting the fastest ray (see lemmas).  Compute ToF for each possible ray  Select the ray with the smallest ToF. 

Update the lower or the upper bound, 








Update loop parameters, 


Compute the estimated horizontal distance between the nodes.

The Cramér–Rao bound (CRB) expresses a lower bound on the variance of any unbiased estimator of a deterministic parameter. As mentioned before, since the depth information is known and the projection method can be used for localization, a given distancebased traditional localization algorithm works only with horizontal distances. Therefore, in this section we only derive the CRB for the horizontal distance estimation between two nodes. For the computation of the horizontal distance, three measurements are required: two depth measurements which are not directly related to the horizontal distance, and one ToF measurement. It is assumed that all the measurements are affected by Gaussian distributed noise as
In order to compute the partial derivative
Note that the above conversion can only be done after we compute the fastest ray, because only then we are able to locate the maximum and/or minimum points on the trajectory and build the new environment. Under this assumption, and using
In this section we study the performance of finding the fastest path, as well as the proposed ranging algorithm in a multilayer underwater environment [
In this part of the report, based on the aforementioned lemmas, we analyze how a ray can propagate between two points inside the shallow water medium. Using the
In
In
In
In
We have analyzed the problem of localizing a target node in an underwater environment. The inhomogeneous underwater medium upsets the linear dependency of the pairwise distances to the time of flight. We have shown that, if the depth information of the unlocalized node is available, then the problem of underwater localization can be converted to the traditional rangebased one. Dividing the underwater medium into several isogradient sound speed profile layers, we have completely analyzed how a ray can travel between two given points through using different Lemmas. Further, we have proposed an iterative algorithm for the range estimation between two nodes, and we have demonstrated that the proposed algorithm attains the CRB and performs superb in comparison with other existing algorithms. In the future, we want to extend this work for more elaborate SSPs (not necessarily multiple isogradient), especially the ones with one local minimum, for ranging and channel modeling applications.
The research leading to these results has received funding from the European Commission FP7ICT Cognitive Systems, Interaction, and Robotics under the contract #270180 (NOPTILUS).
Projection of pairwise distances on the horizontal plane crossing the target.
Samples of ray trajectories as they travel through different layers.
ToF error of the straightline propagation model in a single layer for different values of range and depth.
Linear dependency of the reflection and crossing points under the assumption of a perfect reflection.
Changing the real ray trajectory into a trajectory which is a monotonic function of the depth.
Sound speed profile for deep and shallow water.
Sample of ray propagation between two nodes.
Different possible rays between two points in the second layer.
Performance of the proposed algorithm for deep water.
Performance of the proposed algorithm for difference values of noise power.
All possible patterns that a fastest ray in a shallow underwater environment can follow.
from layer 1  1  1.2  1.2.3 
1.1.2  1.1.2.3  
1.2.3.2  1.2.3.3  
1.2.3.3.2  1.2.3.2.3  
 
from layer 2  2.1  2  2.3 
2.1.1  2.1.1.2  2.1.1.2.3  
2.3.2  2.3.3  
2.3.2.3.2  2.3.2.3  
⋮  ⋮  
 
from layer 3  3.2.1  3.2  3 
3.3.2.1  3.2.1.1.2  3.3  
3.3.2.1.1  3.2.3.2  3.2.3  
3.2.3.3.2  3.2.1.1.2.3  
3.2.3.2.3.2  3.2.3.2.3  
⋮  ⋮ 