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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In an underwater acoustic channel, the propagation conditions are known to vary in time, causing the deviation of the received signal strength from the nominal value predicted by a deterministic propagation model. To facilitate a large-scale system design in such conditions (e.g., power allocation), we have developed a statistical propagation model in which the transmission loss is treated as a random variable. By applying repetitive computation to the acoustic field, using ray tracing for a set of varying environmental conditions (surface height, wave activity, small node displacements around nominal locations,

The growing need for ocean observation and remote sensing has recently motivated a surge of research publications as well as several experimental efforts (e.g., [

This fact has been gaining recognition in the research community, leading to increased awareness about the need for network simulators that take into account the physics of acoustic propagation [

While it is well known from field experiments that the received power varies in time around the nominal value predicted by a deterministic propagation model, little is known about the statistical nature of these variations. Literature on this topic is scarce; however, several recent references indicate that the received signal strength obeys a log-normal distribution (e.g., [

In this work, we analyze those random variations in the large-scale transmission loss that are mainly governed by environmental factors, such as surface activity (waves) for a particular network scenario. We begin by employing a prediction model based on the Bellhop ray tracing tool [

While it is possible, in principle, to run a deterministic propagation model for a large number of different surface conditions, the underlying computational demands are high. In a large network, it is ineffective, and possibly not even feasible, to run a complex prediction model for each packet transmission. A statistical prediction model then becomes necessary.

The goal of our work is to employ an existing deterministic prediction model (DPM), such as the ray tracer [

Whereas repeated computation of the ray trace for different hops that each of the data packets traverses in a given network may be computationally prohibitive, statistical modeling requires only a single call to the Gaussian random generator for each packet transmission. Thus, the overall simulation time is considerably reduced, allowing a system designer to freely experiment with different network protocols and resource allocation strategies in an efficient manner. So, the ultimate goal is to choose the best upper-layer protocol suite and to relate the necessary system resources (power, bandwidth) to the propagation conditions,

Tradeoff between model complexity and accuracy is shown in

The rest of this work is organized as follows: in Section 2 we define a specific network scenario and discuss the computational demands of deterministic propagation models. The statistical propagation model we propose is described in Section 3. In Section 4 we discuss the implications that statistical modeling can have on network planning. Finally, in Section 5 some conclusions are drawn.

Now we are going to define the overall system where we have developed our study. First, we will define the geographical location and dimensions of the network scenario, including the environmental parameters like bathymetry, floor sediment composition, sound speed profile, water temperature and surface wave activity, among others, that could be found in global ocean databases [

The network of interest is located in coastal waters near Valencia, Spain, at coordinates 39°48′13.14″N and 0°4′34.53″W. It consists of eight nodes arranged in a linear topology, as illustrated in

For our purposes, the source is assumed to be at one end (closest node to the coastline), and the rest of the nodes are placed at different distances ranging from 500 m to 3,700 m. All nodes are anchored to the sea floor in such a way that their depth is 10 meters, while the water depth varies from 25 m to 35 m within the network scenario of 5,000 × 5,000 m^{2}. Should we wish to employ a different network scenario, the procedure would be the same, since all network scenario and environmental parameters could be obtained from on-line global databases, and the rest of parameters may be fixed in our simulation framework. We assume a fixed network topology, and vary the parameters related to the surface wave activity (wave height and wave length). The surface parameters are taken from historical and prediction values from National Geophysical Data Center databases [

We also account for the fact that an acoustic communication signal does not consist of a single frequency, but occupies a (possibly wide) certain bandwidth as a result of the acoustic signal modulation employed. The overall transmission loss is computed along the whole network scenario by running the DPM with the Bellhop approach. Each DPM simulation run produces the acoustic field values in a 5 km × 5 km × 30 m volume, with a resolution of 0.33 m^{3}. The values corresponding to receiver node locations are then extracted, and a statistical analysis is performed for each position.

To compute the transmission loss, we have used two different approaches: (1) assuming single frequency acoustic signals, where several experiments were performed with frequency ranging from 5 to 80 kHz; and (2) assuming a more realistic approach, taking as reference the Evologics Modems technical data sheet [

Although the network topology is fixed,

For each experiment, all network nodes employ the same power transmission.

The hardware used to run all the simulations is a cluster of computers that consist of six nodes, each one with four CPUs of 1 GHz and 8 GB of RAM, for a total of 24 cores, all governed by Rocks Cluster OS version 4.3 [

Each execution of the Bellhop tool [

We have introduced the fact that an ensemble of transmission loss values, obtained by varying the physical conditions along a range of frequencies, obeys a log-normal distribution. The statistical model proposal is an attempt to replace this heavy computational process with a simple expression that offers transmission loss predictions as reliable as the propagation model. The study of the log-normal distribution requires focusing on both the parameters required to build the distribution, the mean (

The study of the mean value of the

In order to estimate the mean transmission loss, we have employed the Surface Fitting Tool from MATLAB R2011a [

Achieving a coefficient of determination (R2) of 0.96, the single frequency mean transmission loss, sfμ(d,f), is obtained with the resulting fitting _{1}_{2}_{3}_{4}_{5}

In

Since real implementations perform signal modulations that produce a specific bandwidth, not a single frequency, we proceed to extend the single frequency SPM model defined above to acoustic signals with a particular bandwidth. The process to obtain the average transmission loss (signal attenuation values) out of a range of frequencies is done as follows: for each spatial position in the network scenario, we calculate the inverse of the attenuation values for each single frequency composing the desired bandwidth, and then we obtain their average. In

Now, the transmission loss corresponding with the three frequency bands are plotted together with the transmission loss of their central frequencies calculated with

_{6}_{7}_{8}_{9}_{10}_{11}_{12}

Applying the bandwidth correction factor to the SPM single frequency approach, the estimation error is considerably reduced, as it can be seen at

It is time now to analyze the impact on the transmission loss of the node movement defined in our target scenario, where three different node movement models have been defined. These movement models are parameterized with depth and range with a bandwidth signal of 5–15 kHz (center frequency 10 and bandwidth 11 kHz) at every node position in the scenario.

In

The study of the Standard Deviation Value (σ) is essential to obtain a statistical expression that would accurately describe the behavior shown in

In order to study σ, we will use the same bandwidth signals as the ones used in the previous section, 5–15 kHz, 20–34 kHz and 50–75 kHz, as well as the same node movement model described in

As it can be seen at _{1}_{2}_{3}_{4}_{5}_{6}_{7}_{8}_{9} = 2.013.

Finally, we have determined the mean and standard deviation values of a log-normal distribution that properly represents the same behavior as the Bellhop acoustic propagation model, taking into account the transmission loss variability induced by environmental scenario parameters, and the node movement typically found in underwater deployments.

The apparent match between the results of deterministic and statistical models motivates the SPM use for network design and analysis via simulation. Consider, for example, network simulation over a prolonged interval of time that spans varying propagation conditions and involves the transmission of a large number of data packets over multiple hops. If deterministic modeling is used, each packet transmission requires one execution of the Bellhop ray tracer, which soon becomes excessively long for a growing number of data packets (assuming 5 minutes for each Bellhop run and a single frequency, simulation of 100,000 packets would take about a year). Although the DPM offers an accurate solution for the particular geometry observed at any given moment in time, its execution makes the simulation times unaffordable for the benchmarking and testing of the upper layer protocols.

In contrast, a statistical model can take several hours to compute (40 hours in the example we have presented) a particular network scenario, but this would be needed only once for a network scenario. After that, for a particular simulation run, each packet transmission only requires a single call to a Gaussian random number generator to determine the transmission loss. Moreover, if network topology changes slightly, or if a new node is added, the statistical model needs to be augmented only by the corresponding set of nominal parameters (mean and standard deviation for the newly created links).

Most important, the statistical model can easily be used to assess transmit power allocation that will guarantee successful data packet reception with a desired level of performance (e.g., link reliability).That is to say, the proposed SPM can easily be used to calculate the transmission loss values that are not exceeded with a given probability. For example, a 90% transmission loss would be that value which is not exceeded for 90% of the time,

We have highlighted three link reliability levels, corresponding to successful channel realization probabilities of 50%, 75%, and 90%. This information will assess the transmission power required to guarantee the destination node reachability with a specific probability.

The availability of X% values is significant for determining the transmit power necessary to achieve a certain level of performance. Typically, network planning is based on the nominal ray trace,

Large-scale design of an underwater acoustic network requires a judicious allocation of the transmit power across different links, to ensure a desired level of system performance (connectivity, throughput, reliability,

While, in principle, it is possible to examine the network performance for a large set of perturbed propagation conditions, the computational complexity involved in doing so is extremely high. To facilitate network simulation in the presence of channel fading, we investigated a statistical modeling approach. Our approach is based on establishing the nominal system parameters for a desired deployment location (water depth, sediment composition, operational frequency range) and using ray tracing to compute an ensemble of transmission losses for typical inter-node distances. An ensemble is generated by considering a set of perturbed surface conditions, defined by varying wave activity (height, period). The so-obtained ensemble is then used to determine the statistical parameters of a hypothesized log-normal distribution of the transmission loss. For a representative example of a small network operating in a 5 km × 5 km area with inter-node distances ranging between 500 m and 4 km, it was found that the mean can be well approximated as a linear function of the logarithm of distance, while the variance can be modeled as constant over given ranges of distances. More elaborate and more accurate models than the lognormal one can also be developed using this approach.

A statistical model of this type enables computationally-efficient inclusion of fading effects into a network simulator. Namely, to assess the average system performance, network operation has to be simulated over a large set of channel realizations (e.g., varying surface conditions). Whereas repeated computation of the ray trace for different hops traversed by each of the data packets in a given network may be computationally prohibitive, statistical modeling requires only a single call to the Gaussian random generator for each packet transmission. The overall simulation time is thus considerably reduced, allowing a system designer to freely experiment with varying protocols and resource allocation strategies in an efficient manner. The ultimate goal of such computational experiments is to choose the best upper-layer protocol suite and relate the necessary system resources (power, bandwidth) to the propagation conditions,

This work was supported by the Ministry of Science and Education of Spain under Projects DPI2007-66796-C03-03 and TIN2011-27543-C03-03.

An ensemble of transmission losses calculated by the Bellhop model. Solid line indicates the average calculated over the total run time. Dashed lines indicate the values of one standard deviation σ.

Tradeoff between model propagation accuracy and computational complexity.

Network deployment in Valencia, Spain.

Network node movement model.

Average transmission loss

Attenuation of bandwidth signals with DPM and SPM single frequency proposal.

Bandwidth signals loss with DPM and SPM single frequency proposal after applying the bandwidth correction factor.

Attenuation

Gateway reachability (central node) from Node #1 (bottom leftmost node).

Standard deviation of attenuation

Transmission loss normalized histogram of an ensemble of channel realizations.

The transmission

System Parameters.

Transmission range | 500 m to 3,700 m (in steps of ∼500 m) |

Scenario Area | 5,000 m × 5,000 m |

Sediment floor | Gravel |

Month | August |

Wave height (m) | 1 m to 3 m (in steps of 0.15 m) |

Wave length (m) | 100 m to 150 m (in steps of 3.5 m) |

Frequency (kHz) | 5 to 80 kHz (in steps of 5 kHz) |

Scenario depth (m) | 25–35 |

Global load (packets/s) | 5 |

Data packet size (bits) | 1,024 |

Control packet size (bits) | 24 |

Simulation time (s) | 3,600 |