On the Relationship Between Capacity and Distance in
an Underwater Acoustic Communication Channel
∗
Milica Stojanovic
millitsa@mit.edu
Massachusetts Institute of Technology, Cambridge, U.S.A.
Path loss of an underwater acoustic communication channel depends not only on the trans-
mission distance, but also on the signal frequency. As a result, the useful bandwidth de-
pends on the transmission distance, a feature that distinguishes an underwater acoustic
system from a terrestrial radio one. This fact influences the design of an acoustic network:
a greater information throughput is available if messages are relayed over multiple short
hops instead of being transmitted directly over one long hop. We asses the bandwidth de-
pendency on the distance using an analytical method that takes into account physical mod-
els of acoustic propagation loss and ambient noise. A simple, single-path time-invariant
model is considered as a first step. To assess the fundamental bandwidth limitation, we
take an information-theoretic approach and define the bandwidth corresponding to optimal
signal energy allocation – one that maximizes the channel capacity subject to the constraint
that the transmission power is finite. Numerical evaluation quantifies the bandwidth and
the channel capacity, as well as the transmission power needed to achieve a pre-specified
SNR threshold, as functions of distance. These results lead to closed-form approximations,
which may become useful tools in the design and analysis of acoustic networks.
I. Introduction
With the availability of high speed acoustic commu-
nication techniques, the maturing of underwater ve-
hicles, and the advances in sensor technology, in-
tegration of point-to-point communication links into
autonomous underwater networks has been steadily
gaining interest over the past years, both from the re-
search viewpoint [1], and that of the design and de-
ployment of first experimental networks [2]. It is en-
visioned that some of the immediate applications of
acoustic networking technology will include collabo-
rative missions of multiple autonomous vehicles, and
the deployment of ad hoc underwater sensor networks.
∗
This material is based upon work supported by the National
Science Foundation under Grant No. 0520075.
c
ACM, 2007. This is a revision of the work pub-
lished in the Proceedings of the First ACM International
Workshop on Underwater Networks (WUWNet) 2006,
http://doi.acm.org/10.1145/1161039.1161049.
Permission to make digital or hard copies of all or part of
this work for personal or classroom use is granted without
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WUWNet’06, September 25, 2006, Los Angeles, Califor-
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Copyright 2006 ACM 1-59593-484-7/06/0009 ...$5.00
The design of such systems is the subject of on-going
research.
One of the questions that arise naturally at this time
is what are the fundamental capabilities of underwa-
ter networks in supporting multiple users that wish
to communicate to (or through) each other over an
acoustic channel. While research has been extremely
active on assessing the capacity of wireless radio net-
works (e.g., [3]) no similar analyses have been re-
ported for underwater acoustic networks. The few
available analyses focus on the acoustic channel ca-
pacity. For example, [4] uses a time-invariant channel
model with additive Gaussian noise that may or may
not be white, while [5] uses a Rayleigh fading model,
with additive white Gaussian noise (AWGN). Neither
of these analyses addresses the capacity dependence
on distance.
Underwater acoustic communication channels are
characterized by a path loss that depends not only on
the distance between the transmitter and receiver, as it
is the case in many other wireless channels, but also
on the signal frequency. The signal frequency deter-
mines the absorption loss which occurs because of the
transfer of acoustic energy into heat. This fact implies
the dependence of acoustic bandwidth on the com-
munication distance. The resulting bandwidth limi-
tation is a fundamental one, as it is determined by the
physics of acoustic propagation, and not by the con-
straints of transducers.
The absorption loss increases with frequency as
well as with distance, eventually imposing a limit
on the available bandwidth within the practical con-
straints of finite transmission power. Consequently,
a shorter communication link offers more bandwidth
than a longer one in an underwater acoustic system.
For example, transmission over 100 km can be per-
formed in one hop, using a bandwidth of 1 kHz, or by
relaying the information over 10 hops, each of which
is 10 km long, but offers a bandwidth on the order of
10 kHz. Hence, in exchange for a more complicated
system of relays, significant increase in information
throughput can be obtained. At the same time, total
energy consumption will be lower, but this is so for
the radio channel as well.
Before one can answer the questions of network
capacity, a functional dependence of the acoustic
communication bandwidth with distance must be ob-
tained. This is the subject of the present paper, which
is organized as follows.
In Sec.II we summarize the basics of acoustic prop-
agation, to formulate a model of the path loss and the
ambient noise that will be used to assess the band-
width. In Sec.III we propose two definitions of the
acoustic bandwidth, one a heuristic definition based
on the 3 dB loss in the band-edge SNR and a uni-
form energy allocation, and the other an information-
theoretic definition based on optimal energy alloca-
tion for a fixed transmission power. In both cases, the
total transmission power is determined as that needed
to achieve a pre-specified SNR within the given band-
width. Sec.IV illustrates the results numerically, pro-
viding a quantitativemeasures of the bandwidth in Hz
and capacity in bps, as well as the transmission power
in dB re μ Pa, as functions of distance. Numerical re-
sults lead to closed-form approximations which pro-
vide functional dependence of the system capacity on
the transmission distance. Conclusions are summa-
rized in Sec.V.
II. Acoustic propagation: path loss
and noise
II.A. Attenuation
Attenuation, or path loss that occurs in an underwa-
ter acoustic channel over a distance l for a signal of
frequency f is given by
A(l, f)=A
0
l
k
a(f)
l
(1)
where A
0
is a unit-normalizing constant, k is the
spreading factor, and a(f) is the absorption coeffi-
cient. Expressed in dB, the acoustic path loss is given
by
10 log A(l, f)/A
0
= k · 10 logl + l · 10 log a(f) (2)
The first term in the above summation represents the
spreading loss, and the second term represents the ab-
sorption loss. The spreading factor k describes the ge-
ometry of propagation, and its commonly used values
are k =2for spherical spreading, k =1for cylindri-
cal spreading, and k =1.5 for the so-called practical
spreading. (The counterpart of k in a radio channel is
the path loss exponentwhose value is usually between
2 and 4, the former representing free-space line-of-
sight propagation, and the latter representing two-ray
ground-reflection model.) The absorption coefficient
can be expressed empirically, using the Thorp’s for-
mula which gives a(f) in dB/km for f in kHz as [6]:
10 log a(f)=0.11
f
2
1+f
2
+44
f
2
4100 + f
2
+
2.75 · 10
−4
f
2
+0.003 (3)
This formula is generally validfor frequencies above a
few hundred Hz. For lower frequencies, the following
formula may be used:
10 log a(f)=0.002 + 0.11
f
2
1+f
2
+0.011f
2
(4)
The absorption coefficient is shown in Fig.1. It in-
creases rapidly with frequency, thus imposing a limit
on the maximal usable frequency for an acoustic link
of a given distance.
0 100 200 300 400 500 600 700 800 900 1000
0
50
100
150
200
250
300
350
frequency [kHz]
absorption coefficient [dB/km]
Figure 1: Absorption coefficient, a(f) [dB/km].
The path loss describes the attenuation on a sin-
gle, unobstructed propagation path. If a tone of fre-
quency f and power P is transmitted over this path,
the received signal power will be P/A(l, f). If there
are multiple propagation paths, each of length l
p
,p=
0,...P − 1, then the channel transfer function can be
described by
H(l, f)=
P −1
p=0
Γ
p
/
A(l
p
,f)e
−j2πfτ
p
(5)
where l = l
0
is the distance between the transmitter
and receiver, Γ
p
models additional losses incurred on
the pth path (e.g. reflection loss), and τ
p
= l
p
/c is the
delay (c=1500 m/s is the nominal speed of sound un-
derwater). If the transmission is not directional, such
that propagation paths other than the direct one con-
tribute to the received signal, then the received power
will be P |H(l, f)|
2
. In our treatment, we shall fo-
cus on a propagation model that takes into account
only the basic path loss. Extensions to the multipath
propagation case are straightforward, if the attenua-
tion A(l, f) is replaced by 1/|H(l, f)|
2
, evaluated for
the particular channel geometry or determined exper-
imentally.
II.B. Noise
The ambient noise in the ocean can be modeled using
four sources: turbulence, shipping, waves, and ther-
mal noise. Most of the ambient noise sources can
be described by Gaussian statistics and a continuous
power spectral density (p.s.d.). The following empir-
ical formulae give the p.s.d. of the four noise compo-
nents in dB re μ Pa per Hz as a function of frequency
in kHz [7]:
10 log N
t
(f)=17− 30 log f
10 log N
s
(f) = 40 + 20(s − 0.5) + 26 log f −
60 log(f +0.03)
10 log N
w
(f)=50+7.5w
1/2
+20logf −
40 log(f +0.4)
10 log N
th
(f)=−15 + 20 logf (6)
Turbulence noise influences only the very low fre-
quency region, f<10 Hz. Noise caused by distant
shipping is dominant in the frequency region 10 Hz
-100 Hz, and it is modeled through the shipping activ-
ity factor s, whose value ranges between 0 and 1 for
low and high activity, respectively. Surface motion,
caused by wind-driven waves (w is the wind speed in
m/s) is the major factor contributing to the noise in
the frequency region 100 Hz - 100 kHz (which is the
operating region used by the majority of acoustic sys-
tems). Finally, thermal noise becomes dominant for
f>100 kHz.
The overall p.s.d. of the ambient noise, N(f)=
N
t
(f)+N
s
(f)+N
w
(f)+N
th
(f), is illustrated in
Fig.2, for the cases of no wind (solid) and wind at
a moderate 10 m/s (dotted), with varying degrees of
shipping activity in each case. The noise decays with
frequency, thus limiting the useful acoustic bandwidth
from below. It may be useful to note that in a certain
frequency region the noise p.s.d. decays linearly on
the logarithmic scale. The following approximation
may then be useful:
10 log N(f) ≈ N
1
− η log f (7)
This approximation is shown in the figure (dash-dot)
with N
1
=50dB re μ Pa and η=18 dB/decade.
10
0
10
1
10
2
10
3
10
4
10
5
10
6
20
30
40
50
60
70
80
90
100
110
f [Hz]
noise p.s.d. [dB re micro Pa]
........ wind at 10 m/s
____ wind at 0 m/s
shipping activity 0, 0.5 and 1
(bottom to top)
Figure 2: Power spectral density of the ambient noise,
N(f) [dB re μ Pa]. The dash-dot line shows an ap-
proximation 10 log N (f)=50− 18logf.
II.C. The AN Product and the SNR
Using the attenuation A(l, f) and the noise p.s.d.
N(f) one can evaluate the signal-to-noiseratio (SNR)
observed over a distance l when the transmitted signal
is a tone of frequency f and power P . Not count-
ing the directivity gains and losses other than the path
loss, the narrow-band SNR is given by
SNR(l, f)=
P/A(l, f)
N(f)Δf
(8)
where Δf is the receiver noise bandwidth (a nar-
row band around the frequency f). The AN product,
A(l, f)N (f), determines the frequency-dependent
part of the SNR. The factor 1/A(l, f)N (f) is illus-
trated in Fig.3. For each transmission distance l, there
clearly existsan optimal frequency f
o
(l) for which the
maximal narrow-band SNR is obtained. The optimal
frequency is plotted in Fig.4 as a function of trans-
mission distance. In practice, one may choose some
transmission bandwidth around f
o
(l), and adjust the
transmission power so as to achieve the desired SNR
level. We comment more on such choices in the fol-
lowing section.
0 2 4 6 8 10 12 14 16 18 2
0
−170
−160
−150
−140
−130
−120
−110
−100
−90
−80
−70
5km
10km
50km
100km
frequency [kHz]
1/AN [dB]
Figure 3: Frequency-dependent part of narrow-band
SNR, 1/A(l, f)N (f). Practical spreading (k =1.5)
is used for the path loss A(l, f). Moderate shipping
activity (s =0.5) and no wind (w =0) are used for
the noise p.s.d. N (f).
0 10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
35
distance [km]
optimal frequency [kHz]
Figure 4: Optimal frequency f
o
(l) is the one at which
1/A(l, f)N (f) reaches its maximum.
III. Bandwidth and capacity
III.A. A heuristic bandwidth definition
A possible definition of the system bandwidth is that
of a 3 db (or some other level) bandwidth. We
define the 3 dB bandwidth B
3
(l) as that range of
frequencies around f
o
(l) for which SNR(l, f) >
SNR(l, f
o
(l))/2, i.e., for which A(l, f)N (f) <
2A(l, f
o
(l))N (f
o
(l)) = 2AN
min
(l).
Once the transmission bandwidth is set to some
B(l)=[f
min
(l),
f
max
(l)] around f
o
(l), the transmission power P (l)
can be adjusted to achieve the desired narrow-band
SNR level at f
o
(l). Alternatively, and perhaps more
meaningfully, one may set the desired transmission
power in accordance with the total SNR correspond-
ing to the bandwidth B(l). If we denote by S
l
(f) the
p.s.d. of the transmitted signal chosen for the distance
l, then the total transmitted power is
P (l)=
B(l)
S
l
(f)df (9)
and the SNR is
SNR(l, B(l)) =
B(l)
S
l
(f)A
−1
(l, f)df
B(l)
N(f)df
(10)
In this definition, the SNR depends on the transmit-
ted signal p.s.d., and so does the total transmission
power P (l). In the simplest case, the transmitted sig-
nal p.s.d. is flat, S(l, f)=S
l
for f ∈ B(l), and
0 elsewhere. The total transmission power is then
P (l)=S
l
B(l). If it is required that the received SNR
be greater than some pre-specified threshold SNR
0
,
then the minimal transmission power can be deter-
mined from SNR
0
and B(l). When the 3 dB band-
width is used, the corresponding transmission power
is determined as
P
3
(l)=SNR
0
B
3
(l)
B
3
(l)
N(f)df
B
3
(l)
A
−1
(l, f)df
(11)
While this definition of the acoustic system band-
width may be intuitivelysatisfying, there is nothing to
guarantee its optimality. It may be possible to achieve
a better utilization of resources through a different
energy distribution across the system bandwidth. In
other words, we may adjust the signal p.s.d. S
l
(f)
in accordance with the given channel and noise char-
acteristics A(l, f) and N(f) so as to optimize some
performance metric. We do so in the following sec-
tion.
III.B. Capacity-based bandwidth defini-
tion
A performance metric that naturally comes to mind is
the channel capacity. Assumingthat the noise is Gaus-
sian, and that the channel is time-invariant for some
intervalof time, the capacity can be obtained by divid-
ing the total bandwidth into many narrow sub-bands,
and summing the individual capacities. The ith sub-
band is centered around frequency f
i
,i=1, 2,...and
it has width Δf, which is small enough that the chan-
nel transfer function appears frequency-nonselective,
i.e. the only distortion comes from a constant attenua-
tion factor A(l, f
i
). The noise in this narrow sub-band
can be approximated as white, with the p.s.d. N(f
i
),
and the resulting capacity is given by
C(l)=
i
Δf log
2
1+
S
l
(f
i
)A
−1
(l, f
i
)
N(f
i
)
(12)
Maximizing the capacity with respect to S
l
(f), sub-
ject to the constraint that the total transmitted power
P (l) is finite, yields the optimal energy distribution.
The signal p.s.d. should satisfy the water-filling prin-
ciple [8]:
S
l
(f)+A(l, f)N (f)=K
l
(13)
where K
l
is a constant whose value is to be deter-
mined from the power P (l), and it is understood that
S
l
(f) ≥ 0.
The power P (l) can be chosen to provide a desired
SNR, SNR
0
, similarly as before. The SNR corre-
sponding to the optimal energy distribution is given
by
SNR(l, B(l)) =
B(l)
S
l
(f)A
−1
(l, f)df
B(l)
N(f)df
= K
l
B(l)
A
−1
(l, f)df
B(l)
N(f)df
− 1(14)
The transmitted power is
P (l)=
B(l)
S
l
(f)df = K
l
B(l)−
B(l)
A(l, f)N (f)df
(15)
If the power is determined as the minimum needed
to satisfy the SNR condition
SNR(l, B(l)) ≥ SNR
0
(16)
then the optimal energy distribution S
l
(f) can be ob-
tained through the following numerical procedure.
For each distance l, we begin by finding the opti-
mal frequency f
o
(l), and setting the initial value of the
constant K
l
to K
(0)
l
= AN
min
(l). We then proceed
iteratively, increasing K
l
in each step by some small
amount, until the condition (16) is met. In particular,
if K
(n)
l
denotes the current value of the constant K
l
,
for which the SNR is still below the desired threshold,
then the following operations are performed in the n-
th step:
1. Determine B
(n)
(l) as that region of frequencies
for which A(l, f)N (f) ≤ K
(n)
l
.
2. Calculate SNR
(n)
from (14) using the band-
width B
(n)
(l) and the constant K
(n)
l
.
3. Compare SNR
(n)
to SNR
0
.IfSNR
(n)
<
SNR
0
, increase K
l
by a small amount, and con-
tinue the procedure. For example, K
(n+1)
l
=
(1 + )K
(n)
l
was used for numerical evaluation
of results in Sec. IV, with =0.01.
When SNR
(n)
reaches (or slightly exceeds) SNR
0
,
the procedure ends. The current value of K
(n)
l
is set
as the desired constant K
l
, and the current value of
the bandwidth B
(n)
(l) is set as the desired bandwidth
B(l). The optimal energy distribution is
S
l
(f)=
K
l
− A(l, f)N (f),f∈ B(l)
0, otherwise
(17)
and the total power is obtained from (15). Finally, the
channel capacity is
C(l)=
B(l)
log
2
K
l
A(l, f)N (f)
df (18)
In comparison, the capacity (if it may be called that)
of the heuristic scheme that uses equal energy distri-
bution across the 3 dB bandwidth is
C
3
(l)=
B
3
(l)
log
2
1+
P
3
(l)/B
3
(l)
A(l, f)N (f)
df (19)
IV. Numerical results
The bandwidth, capacity, and transmission power
were evaluated through numerical integration of the
expressions presented in the previous section. Re-
sults are presented for both the 3 dB definition and
the capacity-maximizing definition of bandwidth. For
lack of better names, we shall refer to these two cases
at the heuristic case and the optimal case, respectively.
In both cases, the acoustic loss is modeled using prac-
tical spreading, k =1.5, and the noise p.s.d. is that
obtained for moderate shipping activity s =0.5 and
![Figure 1: Absorption coefficient, a(f) [dB/km].](/figures/figure-1-absorption-coefficient-a-f-db-km-2wfzx1j6.webp)
![Figure 2: Power spectral density of the ambient noise, N (f) [dB re μ Pa]. The dash-dot line shows an approximation 10 logN (f) = 50− 18logf .](/figures/figure-2-power-spectral-density-of-the-ambient-noise-n-f-db-eeeelmvb.webp)
