Introduction to Acoustics

time as a result of multiple paths, in which different
paths have different arrival times (Fig. 5.55). Hence the
arrival times are related to the acoustic sampling of the
medium. In contrast, standard medical X-ray tomography
utilizes the different attenuation of the paths rather
than arrival time for the inversion process. Since the
ocean sound speed is a function of temperature and
other oceanographic parameters, the arrival structure is
ultimately related to a map of these oceanographic parameters.
Thus, measuring the fluctuations of the arrival
times through the experiments theoretically leads to the
knowledge of the spatial-temporal scale of the ocean
fluctuations.
Tomographic inversion in the ocean has typically
relied on these three points. First, only the arrival
time (and not the amplitude) of the multipath structure
is used as an observable (Table 5.2) [5.67].
Enhanced time-of-arrival resolution is typically obtained
using pulse-compression techniques [5.68] as
mentioned in the bottom-mapping section above. Depending
on the experimental configuration, a choice of
compression is to be made between M-sequences [5.69],
which are strongly Doppler sensitive but have low
temporal side-lobes and frequency-modulated chirps
that are Doppler insensitive with higher side-lobes
(Fig. 5.43). Second, the inversion is performed by comparing
the experimental arrival times to those given by
a model (Fig. 5.56). Last, tomographic inversion algorithm
classically deals with a linearized problem. This
means that the model has to match the experimental
data so that the inversion only deals with small
perturbations.
Thus, ocean tomography starts from a sound-speed
profile c(r) on which small perturbations are added
δc(r, t) ≪ c. The ocean model c(r) has to be accurate
enough to relate without ambiguity an experimentally
measured travel time Ti to a model-deduced ray path
Γi. Typically, some baseline oceanographic information
is known so that one searches for departures from this
baseline information. The perturbation infers a change
of travel time δTi along the ray such that, in a first linear
approximation
�
δTi ≈
Γi
−δc
ds , (5.94)
c2 where Γi correspond to the Fermat path of the unperturbed
ray. An efficient implementation of the inversion
procedure utilizes a projection of the local sound-speed
fluctuations δc(r, t) on a set of chosen functions Ψk(r)
that constitutes a basis of the ocean structure. We have
Underwater Acoustics 5.7 Active SONAR Processing 193
Channel 1 Channel 2
Fig. 5.52 Experimental examples of eight quadrature amplitude
modulation (QAM) transmissions in a multiple-input multipleoutput
configuration (MIMO)at3.5 kHz in a 9 km-long, 120 m-deep
shallow-water ocean. The SNR is 30 dB on channel 1 and 2, the symbol
duration is 1 ms (data rate is 8 kB/s per channel) and bit error
rate (BER) is1×10 −4 (Courtesy H.C. Song, Scripps Institution of
Oceanography)
then
δc(r, t) =
N�
pk(t)ψk(r) , (5.95)
k=1
where pk(t) is a set of unknown parameters. In its most
primitive form, the ocean can be discretized into elementary
cells, each cell being characterized by an unknown
sound-speed perturbation pk. Combining the two above
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One bit
t t
Fig. 5.53 A coherent digital communication system must deal with
the intersymbol interference caused by dispersive multipath environment
of the ocean waveguide (top right). When a sequence of
phase-shifted symbols (in black) are sent, the resulting transmission
(in brown) is fading out because of symbol interference
Part A 5.7