Examining Transmission Power in Minimum Capacity Underwater Acoustic Networks by Kathryn E. Stanchak SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE IN ENGINEERING AS RECOMMENDED BY THE DEPARTMENT OF MECHANICAL ENGINEERING AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARCHES FEBRUARY 2010 0 2009 Massachusetts Institute of Technology All rights reserved . l ) )- OF TECHNOLOGv JUN 3 0 2010 LIBRARIES Signature of Author: DepaAment o echanical Engineering December 18, 2009 Certified By: Franz Hover Doherty Assistant Professor in Ocean Utilization Thesis Supervisor Accepted By: ....... John H. Lienhard V --... ieomns rroiessor i Mvechanical Engineering Chairman, Undergraduate Thesis Committee .... .......... . .......
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Examining Transmission Power in Minimum Capacity UnderwaterAcoustic Networks
by
Kathryn E.Stanchak
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING INPARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
BACHELOR OF SCIENCE IN ENGINEERINGAS RECOMMENDED BY THE DEPARTMENT OF MECHANICAL ENGINEERING
AT THEMASSACHUSETTS INSTITUTE OF TECHNOLOGY ARCHES
FEBRUARY 2010
0 2009 Massachusetts Institute of TechnologyAll rights reserved
. l ) )-
OF TECHNOLOGv
JUN 3 0 2010
LIBRARIES
Signature of Author:DepaAment o echanical Engineering
December 18, 2009
Certified By:Franz Hover
Doherty Assistant Professor in Ocean UtilizationThesis Supervisor
Accepted By: .......John H. Lienhard V
--... ieomns rroiessor i Mvechanical EngineeringChairman, Undergraduate Thesis Committee
.... .......... . .......
Examining Transmission Power in Minimum Capacity UnderwaterAcoustic Networks
by
Kathryn E. Stanchak
Submitted to the Department of Mechanical Engineering onDecember 18, 2009 in Partial Fulfillment of the
Requirements for the Degree of Bachelor of Science inEngineering as Recommended by the
Department of Mechanical Engineering
ABSTRACT
This paper explores the prospect of reducing the transmission power required to operate linkswithin an underwater acoustic network by minimizing the total capacity of the network whilemaintaining certain data flow requirements. This is motivated by an approximate model forunderwater acoustic transmission power that demonstrates that decreasing the distance betweennodes, the capacity of a link between nodes, or both, reduces the power required to send a signalbetween those nodes. A procedure for determining a minimum-sum capacity network developedby Gomory and Hu in 1961 is applied to several common network topologies including tree,ring, and mesh structures. The approximate model for transmission power, which takes intoaccount the large effects of signal attenuation and noise, is used to evaluate these minimalnetworks.
The networks derived from the Gomory-Hu procedure are shown to require less totaltransmission power to operate the entire network. In order to maintain the pre-set data flowrequirements in the Gomory-Hu network, it is necessary to send information across multipleparallel paths in the network. Results show that because of this extra transmission distance, thenetworks derived via the Gomory-Hu procedure and their consequent parallel routing schemesare less efficient in terms of a single-transmission from one node to another node in the networkthan their counterpart networks that operate via a direct-access method, although thetransmission power requirements per node are reduced. This parallel routing scheme implies thatthe Gomory-Hu networks could be beneficial for multi-cast transmission. Results show thatapplying the Gomory-Hu procedure to networks intended for multi-cast instead of single-casttransmission could be a promising way of increasing the efficiency of the overall network.
Thesis Supervisor: Franz HoverTitle: Doherty Assistant Professor in Ocean Utilization
Underwater acoustic networks consist of sensor nodes that communicate with each other
wirelessly by sending acoustic pressure waves through the medium of sea water. The
applications for these networks include environmental monitoring, establishing communications
with underwater vehicles, and as positioning systems for underwater localization and navigation.
Setting up these underwater sensor networks is often a very challenging task. As the pressure
wave travels through the ocean water it can undergo severe signal attenuation and high levels of
noise, much more than its electromagnetic wave counterpart. This results in signal spreading that
can lead to reverberation and also contributes to a very limited bandwidth availability.
Designing and optimizing networks that require a minimum level of transmission power to
operate is an attractive problem for research because it is often extremely expensive to, for
instance, change a battery on a sensor network node located in the deep sea. The focus of this
paper is to analyze power usage in a number of different network layouts. This is done by first
defining the network mathematically and then developing a model for the transmission power
required to transmit underwater. It is found that transmission power increases with respect to
both increasing distance between the two communicating nodes and increasing capacity of that
particular link, or the number of bits per second that can be transmitted. Motivated by this
relationship, different ways of manipulating the distance and capacity of links in a network are
examined. A procedure for a minimal sum-capacity network developed by Gomory and Hu in
1961 is looked at in particular.
The paper is organized as follows: Section 1 covers the representation of flow networks as
graphs and outlines the Gomory-Hu minimal capacity network procedure. Section 2 develops a
closed-form model for the transmission power of a link between two nodes in an underwater
network as a function of distance and capacity. Section 3 describes some simple network
topologies and forms of routing schemes applied to underwater acoustic networks. Section 4
applies the Gomory-Hu procedure in particular to several sample networks that are common to
underwater applications. Section 5 concludes the paper and provides suggestions for further
investigation.
Section 1: Graph Networks
The data signal sent from one node to another in an underwater acoustic network can be thought
of and modeled as a "flow" of information between those nodes. The motivation behind this
section is to develop a way to represent flow networks mathematically, in order to compare them
and perform calculations on them.
1.1: Flow Networks and the Max-Flow Min-Cut Theorem
A flow network can be described in terms of graph theory as graph G (J E) where V is the set
of vertices in the graph and E is the set of edges. For Figure 1.1 shown below, V = {u v w x} and
E = t(u,v) (v, w) (w, x) (x, u)}. Each edge is assigned a capacity of c (u, v)0 ; this is the
weight of the edge for this application. A flow between two vertices is represented as
f(u,v) c(u,v) [1].
UG,(u,v)
V c2(vw)
x
Figure 1. 1: A flow network represented as graph G =(V E).
For this paper, only networks that can be represented as connected graphs will be considered.
This means that for every vertex v e V there is a path s -* v - t where s E v is the source of
the flow and t E v is the sink. The graph shown in Figure 1.1 is connected. A fully-connected
network refers to a network in which each node is capable of transmitting to every other node.
All of the networks looked at in this paper are assumed to be capable of fully-connected
transmission. It will also be assumed that each link in every example network presented is bi-
directional. This means that a flow that can be sent from a source node to a sink node in any of
these networks can also be sent in the opposite direction (from the "sink" to the "source").
A cut in the flow network is any partition of the vertices into two disjoint subsets. The weight of
the cut is the sum of the weights of the edges crossing the cut. For example, a cut of G = (VE) is
shown in Figure 1.2. In this case, the two disjoint subsets are V, = [u} and V2 = {v w x} and the
weight of the cut is the sum of c1 and c4.
Uc,(uYv
vc,(u,x)
c2(v'w)
x
C3(WX)W
Figure 1.2: Flow network G = (V E) cut into two disjoint subsets.
In 1956, Ford and Fulkerson devised an algorithm to compute the maximum flow in a network
from a single source to a single sink [2]. In doing so, they also proved the max-flow min-cut
theorem, which states that the maximum flow in a network from s -+ t is equal to the minimum
capacity cut such that s and t are in different partitions of V For c1 = C4 1 and c2 = C= 2, the cut
shown in Figure 1.2 is the minimum cut and its capacity weight, 2, is also the maximum flow
from u to any other vertex in the network.
1.2: Gomory-Hu Minimal Network and Prim's Algorithm
Gomory and Hu used the max-flow min-cut theorem in 1961 as their primary tool for developing
a synthesis procedure for a minimum capacity network [3] [4]. In their procedure, a minimum
capacity network is defined as the synthesized network in which the sum of the capacities in the
network is at a minimum while the network is still able to fulfill certain capacity requirements.
The idea essentially uses capacity as a cost for each link and the procedure seeks to minimize
this overall cost. The max-flow min-cut theorem is used to ensure that the capacity requirements
of the network are met while minimizing the total capacity.
The Gomory-Hu procedure begins with an initial network of vertices connected by links with
assigned capacities. These initial capacities dictate the flow requirements of the network. The
link capacity from one node connected directly to another node defines the total flow between
those nodes that must be maintained in the final network. An example network, N= (V E)
shown below in Figure 1.3 is used to demonstrate the Gomory-Hu procedure throughout this
section:
a 7 f7
b 12 6 104 10 g
C5 9 6 10
9
d 5 e '9 'h
Figure 1.3: Flow network N = (V E), initial network used in Gomory-Hu demonstration.
The dominant requirement tree referred to by Gomory and Hu is more familiarly known as a
maximum spanning tree. A maximum spanning tree of a graph is the tree with the largest sum of
edge weights that connects all of the vertices in the graph. Multiple algorithms have been
developed for determining a minimum spanning tree which can be easily altered for the problem
of a maximum spanning tree. In this application Prim's Algorithm [5] with the required alteration
for a maximum spanning tree was used. The implementation of Prim's Algorithm in Matlab used
in this paper is given in Appendix (A) with annotations. Figure 1.4 below shows a dominant
requirement tree highlighted for network N:
gC
9 109
d e
Figure 1.4: The dominant requirement tree for network N.
The next step in forming the minimal network is to partition the dominant requirement tree, T,
into a set of uniform trees U= [U ... U,}. The first uniform tree, U, contains the same edges as
T, but each edge has a capacity of c(u, v) = m, the minimum capacity in T. For the next tree, U2,
the edges in T of c(u, v) = m are deleted, and the uniform edge capacity is the min(c(u, v) - m).
These steps are continued to create a forest of uniform trees, all with edge capacities of c(u, v) >
0, as shown in Figure 1.5.1-3:
7 f
Figure 1.5.1: The first uniform tree, U1.
a f
3 3
9C
2 32
d 2 h
Figure 1.5.2: The result of deleting edges of c = 7 from T, and subtracting 7 from
the remaining edge capacities.
U2:
2
C
2
22
2d e
U : f
3
9
3
h
Figure 1.5.3: The remaining uniform trees in the forest derived from N.
Then a set of cycles is formed from the uniform trees that each have at least three vertices. Each
edge in a respective cycle is assigned half of the capacity weight of the edges in the
corresponding uniform tree. This is demonstrated in Figure 1.6. These cycles are merged to
obtain the final minimal network as shown in Figure 1.8.
U :4 a
3c
3.5 f
1.5
9
C4a
3
c
f
1.5
g
1.5
Figure 1.6: The set of cycles C = {C1, C2, C3, C4} form from the set of
uniform trees U derived from T
3.5
d 4.5 3.5 h
Figure 1.8: The final Gomory-Hu minimal network derived from N.
11
The cycles must be formed such that the minimum cut in the uniform tree is maintained to ensure
the final network meets the flow requirements of the dominant requirement tree. Following from
that, the minimum cut must not be increased when the cycle is formed to ensure that the final
network is indeed minimal. This requirement on cycle formation comes directly from the max-
flow min-cut theorem. There may be other solutions to the Gomory-Hu procedure for a particular
initial network because a maximum spanning tree of a graph is not necessarily unique and the
cycles made from the forest of uniform trees may be formulated in any way so long as the
minimum cut is maintained. It can be seen from this example that the Gomory-Hu procedure
takes away edges and adds others to the initial network layout. This is why it is important that
each node in the network is capable of transmitting to every other node, or in other words,
capable of fully-connected transmission. The algorithms written for this paper to complete the
steps involved in the Gomory-Hu minimal network procedure along with Prim's Algorithm are
implemented in the Matlab language in Appendix (A).
Section 2: Underwater Acoustic Channel
In order to analyze underwater acoustic sensor networks, it is important to determine the
fundamental capabilities of nodes communicating through the underwater acoustic channel. In
this section, a closed-form model of the transmission power required to send an acoustic signal
between two nodes in a network is developed. Also, the frequency dependent characteristics of
underwater acoustic transmission are explained in order to better understand these effects on the
final model.
2.1: Frequency Dependent Factors
Unlike communication signals sent within the electromagnetic spectrum, acoustic signals are
pressure waves and consequently they experience more frequency dependent distortion. This
distortion is due to signal path loss, or attenuation, as the acoustic wave propagates through sea
water and the ambient noise always present in deep sea waters. Because many of the problems in
designing feasible and efficient underwater acoustic networks are a result of this high level of
distortion, it is important to understand the contributing factors and to accurately characterize
them in any formulation of a channel model.
2.1.]: Signal Attenuation
The attenuation of an acoustic signal can be thought of as a result of two factors - the
geometrical spreading of the sound pressure wave and the absorption of some of the acoustic
energy [6]. This is modeled as:
A (l, f )(l/1,)k a (f
or, expressed in dB:
10 log (A (l, f ))=k 10 log (l/1,)+l 10 log (a (f))
where I is the distance over which the signal is transferred, bejis a reference distance, k is the
spreading factor, and a(f) is the absorption coefficient. The spreading factor, k, describes the
geometry of the propagating acoustic signal. A practical value that will be used here is k = 1.5,which corresponds to an average of spherical and cylindrical spreading. The absorption
coefficient models the acoustic energy that is transformed into heat as the signal propagates
through the channel [7]. It is actually a function of frequency, salinity, temperature, and
hydrostatic pressure, but an empirically determined formula useful for most practical frequency
ranges gives the absorption coefficient solely as a function of frequency:
[1] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein Clifford. Introduction toAlgorithms, Third Edition. The MIT Press. Cambridge, MA, 2009.
[2] Ford, L.R.; Fulkerson, D.R. "Maximal flow through a network." Canadian Journal ofMathematics. 1956, p. 399-404.
[3] Gomory, R.E; Hu, T.C. "Multi-Terminal Network Flows." Journal of the SocietyforIndustrial andApplied Mathematics, Vol. 9, No. 4. Dec.1961, p. 551-570.
[4] Jungnickel, Dieter; Schade, T. Graphs, Networks, & Algorithms. Springer. New York, 2003.
[5] Prim, R.C. "Shortest connection networks and some generalizations." Bell System TechnicalJournal, No. 36. Dec. 1957, p. 1389-1401.
[6] Stojanovic, Milica. "On the Relationship Between Capacity and Distance in an UnderwaterAcoustic Communication Channel." Mobile Computing and Communications Review, Vol. 11,No. 4. October, 2007, p. 41-47.
[7] Brekhovskikh, L.M.; Lysanov, Yu.P. Fundamentals of Ocean Acoustics, Third Edition. AIPPress. 2003, New York.
[9] Lucani, Daniel E.; Medard, Muriel; Stojanovic, Milica. "Underwatre Acoustic Networks:Channel Models and Network Coding Based Lower Bound to Transmission Power forMulticast." IEEE Journal on Selected Areas in Communications, Vol. 26, No. 9. Dec. 2008 p.1708-1719.
[10] Sozer, Ethem M.; Stojanovic, Milica; Proakis, John G. "Underwater Acoustic Networks."IEEE Journal of Oceanic Engineering, Vol. 25. No. 1. Jan, 2000, p. 72-83.
[ 11 ] Eren, Halit. Wireless Sensors and Instruments: Networks, Design, and Applications. Taylor& Francis Group. Boca Raton, FL, 2006.