We start our analysis by considering a ray traveling between two points in adjacent layers. As illustrated in Figure 2, there are many ways for a ray to travel from U to V. An analysis of all possible rays is feasible, and we will later on discuss this. Now, we only focus on a ray which crosses the intermediate boundary only once, and does not propagate into other layers except these two adjacent layers such as the ray from A to B in Figure 2. In this scenario, when the r coordinate of the crossing point, M, is computed, we are able to relate the positions of the two nodes to the ToF. It can be seen that the ray has two parts, one indexed by p = 1, and the other by p + 1 = 2. The ending point of the first part is the starting point of the second part. Thus, the two parts of the ray can be related to each other according to (12) θ 2 S = θ 1 EAnother representation for Equation (12) can be obtained by taking the tangent from both sides of the equation. Using Equations (9e) to (9h), the boundary equation can then be modified to (13) X 2 + Y 2 1 − K 2 = X 1 − Y 1 1 + K 1

For a two-part ray, the combination of Sub-Equations (9b), (9c), and (9d) for each part, together with boundary Equation (13) forms a third-order polynomial root finding problem where the roots represent the possible r coordinates of the crossing point M. Notice that the node positions and the depth of M are known and as a result, the parameters K₁ and K₂ can be computed easily, X₁ (X₂) is inversely related to Y₁ (Y₂), and the only unknown parameter is r^M which determines X₁ and X₂. Since there are at most three roots for a third order polynomial, there are at most three ways for a ray departing at A to reach B (note that we are still assuming that a ray crosses the boundary once, and propagates only in these two adjacent layers). For each of these possible rays the ToF can be calculated as, (14) t = − 1 a 1 [ ln   1 + sin   θ 1 E cos   θ 1 E − ln   1 + sin   θ 1 S cos   θ 1 S ] + − 1 a 2   [ ln   1 + sin   θ 2 E cos   θ 2 E − ln   1 + sin   θ 2 S cos   θ 2 S ]

To simplify the multi-layer analysis, we define the concept of ray pattern. A ray pattern is a set consisting of all possible rays that can travel between two points. For example, a ray pattern of 2.1.1.2 means that, the ray departs the starting point from the second layer, goes to the first layer, hits the surface, and arrives to the second node in the second layer. Therefore, a ray pattern has several properties. First, the number of digits used in the ray pattern indicates the number of single-layer parts a ray consists of. Second, it shows in which layer each part of a ray is located. Third, the reflection from the sea surface and the seabed can easily be modeled by this concept. Using the ray pattern concept, we are able to show how a ray can travel in a given medium, and which pattern may host the fastest ray.

The procedure of ToF computation, as a function of the node positions for a ray which has multiple single-layer parts is the same as the two-part 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 multi-layer underwater area we introduce several lemmas bellow.

Lemma 1: The sound speed in a layer with zero steepness SSP is constant. Thus, the wave propagation inside that layer is along a straight line, and for that reason the parameter α_p for each truncated ray at that layer is zero. In this case, the relation between the ToF and the node positions is modified into (15) t p = 1 b p ( r p E − r p S ) 2 + ( z p E − z p S ) 2

Lemma 2: Rays are bent toward the region where the sound speed is lower.

Proof: If we place Equation (9b) into Equation (9c), for two nodes at different depths we obtain (16) Y p = 0.5 a p   ( r p E − r p S ) b p + 0.5 a p   ( z p E − z p S )The denominator of Equation (16) is the average of the sound speed at the two points and is always positive, and hence using Equation (9f) we understand that the angle α_p has the same sign as a_p. When a_p is larger than zero, the ray angle at the starting point is greater than the ray angle at the end point or any other point along the ray trajectory. Therefore, the ray angle has a tendency to become smaller and to bend up where the sound speed is lower. On the other hand, when a_p is smaller than zero, the ray angle has a tendency to become larger and to bend down where again the sound speed is lower. Indeed, as depicted in Figure 2, within a layer, a ray bends toward the region where the sound speed is lower.

Lemma 3: In a layer, the depth along a truncated ray between two points, S_p and E_p, can exceed the region [ z p S , z p E ] if and only if (iff) the sign of θ p S   θ p E is negative. The excess value can be computed as, (17) Δ z = { a p   z p S + b p a p   cos   θ p S ( 1 − cos   θ p S ) , if   | θ p S | < | θ p E | a p   z p E + b p a p   cos   θ p E   ( 1 − cos   θ p E ) , if   | θ p S | > | θ p E |

Proof: In a single layer, the depth of a node along a given part as a function of the angle can be derived from Equation (6) (18) z = 1 a p   ( cos   θ k 0 − b p )where k 0 = cos   θ p S a p   z p S + b p = cos   θ p E a p   z p E + b p is a positive constant for a given ray. It is obvious that z follows the behavior of cosθ, and its extremum occurs when θ is zero. In other words, the depth of a node on a truncated ray exceeds the region [ z p S ,   z p E ] iff the signs of the angles at the starting and end points differ from each other. As a result, the excess value can be computed as, (19) Δ z = { max θ   z ( θ ) − max   { z p S ,   z p E } if  a p > 0 min θ   z ( θ ) − min   { z p S ,   z p E } if  a p < 0which leads to Equation (17).

In this way, it can be understood that if Δz for a one-part ray in a single layer is so large that a ray part crosses another layer, the assumption of a one-part ray propagation has to be changed into a three-part ray propagation, and the equations have to be reorganized accordingly (see for instance the ray in Figure 2 traveling between points Y and W).

Lemma 4: A ray can travel multiple times between two layers if the SSP has a local minimum between them.

Proof: A ray which is capable of traveling periodically between two layers has both maximum and minimum points of depth on its traveling path, as illustrated in Figure 2 between the points P and Q. Therefore, based on Lemma 2, it must bend from the first layer to second one, and from the second layer to the first one. Since a ray has a tendency to bend toward the low speed region, periodic traveling between two layers happens if the lower speed region is located between the two layers. In addition, the assumption of an isogradient SSP for each layer forces the total SSP to have a local minimum value at the boundary of these two layers.

Lemma 5: Reflections from the seabed and sea surface can be formulated as a boundary equation. In a shallow underwater environment, it is very common that a traveling ray hits the sea surface or the seabed before reaching the end point. Based on the physical properties of the seabed and the sea surface, the reflection can be formulated as a boundary equation. For instance, if we consider a perfect reflection from the seabed or the sea surface, the boundary equations can be obtained as (20) β p + 1 + α p + 1 = − ( β p − α p ) β p + 1 = − β p

Under the assumption of a perfect reflection, the reflected parts in different layers have the same properties as the corresponding non-reflected parts but with an axial symmetry around the line parallel to the z-axis crossing the reflection point as illustrated in Figure 4. Due to this symmetry which is resulted by the cylindrical symmetry of the ray propagation, the r-coordinate of the two crossing points around a reflection point are linearly related to each other, and one can be formulated by the r-coordinate of the other. Therefore, this does not change the degree of the polynomial resulting from such kinds of ray patterns.

Thanks to the piece-wise 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 ray pattern using the above lemmas we can then for every P part ray relate the ToF to the node’s position using Equation (9) for each single-layer part and the following boundary equations (21) X p + 1 + Y p + 1 1 − K p + 1 = X p − Y p 1 + K p ,     for  p = 1   to  P − 1The ToF of any possible P part ray can then be computed as (22) t = ∑ p = 1 P − 1 a p [ ln   1 + sin   θ p E cos   θ p E − ln   1 + sin   θ p S cos   θ p S ]

For instance, as illustrated in Figure 2, four kinds of rays can be predicted between the two points U and V which are described below.

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.

In Figure 5(a), we show how many rays can travel between two points located in an unbounded two-layer underwater medium. Here, we imagine that the two layers have the same steepness but with different signs, e.g., a₁ = −a₂ = 0.1, and consequently the SSP has a minimum value at the boundary of the two layers. The sound speed at the boundary (z = 0) is assumed to be 1,480 m/s. To compute the number of possible rays between two points, we fix the position of x^S, and change the position of x^E to cover a 100 m by 3.5 km area in the vertical plane as depicted in the figure. According to the discussed lemmas, the possible patterns that can propagate between these two points are 1.2, 1.2.1.2, 1.2.(1.2)…, or 1.1, 1.2.1, 1.(2.1)…, depending on where the two points are located. It is shown that as the pair-wise distance between the two nodes which are located close to the boundary of the two layers increases, the number of paths between them increases too. Around the region where SSP has a minimum value, a pattern with lower number of digits has a lower ToF, but a greater overshoot. Therefore, to compute the fastest ray in this region we can start searching with a simple pattern, and check Lemma 3. If Lemma 3 holds, then we can stop, otherwise we should continue with a more complicated pattern.

For the same scenario as described above, Figure 5(b) shows the error due to a linear approximation of the ToF. From the figure, it can be observed that as the distance between the two nodes gets larger, the linear approximation performs worse. In addition, another effect that influences the accuracy of the straight-line propagation is the angle of this line with the horizontal axis. The larger the angle of the straight line with the horizontal axis, the better the accuracy of the straight-line propagation model.

The above analysis indicates that if a UASN is forced to utilize a straight-line propagation model due to any reason (e.g., system complexity) to achieve more accurate localization results in a noise-free 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.

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 root-finding algorithms can also be used, we utilize the simple bisection algorithm for the results in the simulation section.

Algorithm 1 shows the steps of the proposed algorithm. In this algorithm, K and E are the user-defined limits on the stopping criteria that determine when the algorithm exits from the loop, r_low and r_up are the lower and the upper bound, respectively. The algorithm starts by initializing the upper and the lower bound on the range, and then it computes the fastest ToF for the midpoint of the bounds. In order to calculate the fastest ToF, given the depth of the two points, different ray-patterns that may host the fastest ray are formed, and all the rays between the points are found and their corresponding ToFs are computed, i.e., in Algorithm 1, t^[l] represents the ToF of the l-th found ray between the points. Then, among all these ToFs, the smallest one is selected. Next, based on the computed ToF, the lower, the upper, or both bounds are modified accordingly, and the procedure continues until one of the stopping criteria is met. The important factor that influences the complexity of the proposed algorithm is the number of ray patterns that may host the fastest ray. The ray patterns can be built very efficiently using the proposed Lemmas, still one can add more Lemmas (for a specific SPP) to reduce the number of ray patterns that may host the fastest ray between two 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 distance-based 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 (26) t ^ = t + n t z ^ S = z S + n z S z ^ E = z E + n z Ewhere n_t, n z S and n z E are independent Gaussian distributed with variance σ t 2, σ z 2 and σ z 2, respectively. The Fisher information matrix for estimating the horizontal distance (r^E − r^S), z^S, and z^E can be obtained as (27) I ( r E − r S ,   z S ,   z E ) = 1 σ t 2 [ ∂ t ∂ ( r E − r S ) ∂ t ∂ z S ∂ t ∂ z E ]     [ ∂ t ∂ ( r E − r S ) ∂ t ∂ z S ∂ t ∂ z E ] + [ 0 0 0 0 σ z − 2 0 0 0 σ z − 2 ]

In order to compute the partial derivative ∂ t ∂ ( r E − r S ), we modify the environment in such a way that we can compute the horizontal distance as an integral w.r.t. depth. In order to achieve this, we have to convert the horizontal distance and the ToF to monotonic functions of the depth. Therefore, a ray can not have maximum or minimum points on its trajectory w.r.t. the depth. Let us illustrate the proposed idea with an example. Assume that a ray has a maximum point on its trajectory. The ray angle is zero at this maximum point, and after that it changes sign. But, this sign change does not affect Snell’s law, as it is related to the cosine of the ray angle. As a result, we can assume that the ray travels upward instead of downward as depicted in Figure 6, but in a new environment. In this new environment the SSP of each imaginary region must be changed accordingly. For instance, Figure 6 shows that the real SSP is flipped and translated in the first and second imaginary regions, respectively. In other words, the SSPs of the imaginary regions follow the behavior of the modified ray trajectory.

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 Equations (7b), (7b) and (6) we have (28a) t = ∫ z S z E , m 1 c m   ( z )   1 1 − [ k 0   c m ( z ) ] 2 dz ,           0 < k 0 < min z   1 c m ( z ) (28b) r E − r S = ∫ z S z E , m k 0 c m ( z ) 1 − [ k 0 c m   ( z ) ] 2 dzwhere m indicates that the variable is related to the modified environment. The above equations are similar to Equations (25a) and (25b), hence we can utilize the same approach used in [11] to compute the CRB, which results into (29) var ( r ^ E − r ^ S ) ≥ σ t 2   1 k 0 2 + σ z 2   1 − ( k 0 c ( z E ) ) 2 ( k 0 c ( z E ) ) 2 + σ z 2   1 − ( k 0 c ( z S ) ) 2 ( k 0 c ( z S ) ) 2

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 ray pattern concept, we are able to show how a ray can travel in a given medium, and which pattern may host the fastest ray. In Table 1, we show the family of patterns a ray may travel between two points through different layers. Since the depth of each node is known, we can select the proper patterns from the table, and form the corresponding polynomial formulas. By finding the roots of the polynomials, the ToF of each ray can be computed, and the fastest one will be recognized.

In Figure 8, we illustrate the ray propagation in a shallow underwater environment between two given points in the second layer. A ray can depart the first point and reach the second point in several ways; (a) it can directly propagate in the second layer without entering the other layers; (b) it can go to the upper layer, hit the surface and go back to the second layer; (c) it can go to the third layer and go back to the second layer; (d) it can go to the third layer, hit the bottom and go back to the second layer. Among all of these possibilities we choose the ray which has the lowest ToF. It is worth mentioning that the algorithm of [11] can not compute the correct horizontal range for any of the drawn blue-colored rays in Figure 8 except for the first one, since all other ray trajectories are not monotonic functions of depth.

In Figure 9, we show different possible rays that can travel between two points located in the second layer with a horizontal distance of 1,800 m. Based on the formulation, only three ray patterns can exist in this scenario, i.e., 2.3.2, 2.3.3.2, and 2.1.1.2 (here we only consider one reflection from the surface, and only one reflection from the seabed in the existing ray patterns). Since the sound speed has higher values in the first and second layers, the fastest path belongs to the 2.1.1.2 pattern. It can be noted that if the horizontal distance between the two points increases, one ray pattern will be eliminated, namely 2.3.2.

In Figure 10, we compare the performance of the proposed range algorithm with the one introduced in [11], and with the algorithms which approximate the inhomogeneous underwater medium as a homogeneous one with a presumably constant sound speed, i.e., we use a straight-line range computation based on the depth information. In this simulation, we consider Gaussian noise for the ToF and depth measurements with a standard deviation (std) of σ_t = 1 ms and σ_z = 1 m, respectively. In addition, we choose the deep water environment as a communication medium. The communication is between two points from different layers which are located at depth 550 m and 650 m, respectively. The root mean squared error (RMSE) for the horizontal distance estimation is computed by averaging over 1000 Monte Carlo simulations. As illustrated in this figure, the proposed algorithm performs well for all ranges while the algorithm of [11] has no definite solution from a given point as the horizontal range exceeds a given value. Furthermore, the straight-line algorithm degrades as the distance between the points increases.

In Figure 11, we investigate the effect of the measurement noise on the algorithms under consideration. Here the depth of the two nodes is as before, and their horizontal range is fixed at 3 km. The horizontal axis represents how noisy the measured data are. The depth and ToF measurement noise powers are exponentially related to the parameter λ, i.e., σ_z = 10³σ_t = 10^2−λ. It can be seen that, the performance of the proposed ranging algorithm constantly improves and attains the CRB when we increase the measurement accuracy, while the straight-line estimation does not show any improvement after a given noise power.