SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
1.
Relevant oceanography
1.1
Introduction
For acoustical seafloor mapping techniques the ocean is the acoustic propagation
medium. We therefore consider the ocean and its boundaries from the point of view of
a sound wave.
We need the relation between oceanic properties and underwater acoustic
propagation. The most important acoustic variable is the sound speed c. The average
value of c at sea is approximately 1500 m/s.
Knowledge about the spatial and temporal distribution of c in the ocean is required.
Also important is knowledge about the state of the sea surface and the topography and
composition of the seafloor.
Both the density ρ and sound speed c of sea water are a function of temperature T,
salinity S and hydrostatic pressure P. P is nearly proportional to depth z: P [in dbar]
equals z [in m] to a very good approximation.
c in the upper layer of the ocean strongly depends on T.
1.2
Temperature
The global annual mean temperature distribution at the sea surface (and at 1000 m
depth) shows isotherms that are oriented east-west, see figure 1 and 2. This pattern is
largely due to the distribution of the received solar energy, which gives the strong
latitudinal dependence. The pattern is modified by upwelling and major current
systems (e.g. the Gulf Stream).
Figure 1
7
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 2
The temperature T is highly stratified with depth, i.e., isotherms are nearly parallel to
the horizontal plane. Figure 3 below presents annual averages of T in the Atlantic
Ocean.
Figure 3
8
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
1.3
Salinity
Salinity S is expressed in ‘practical salinity units’ (psu) derived from seawater
conductivity units. In most cases psu corresponds to part per thousand. In the open
ocean S varies much less than T. The S-range for 99 % of the ocean is 33 to 37 psu,
whereas the T-range at the ocean surface is –2 to 30 °C.
The subtropical maxima in S are associated with excess of evaporation over
precipitation, see figure 4.
Figure 4
Unlike T, S does not show consistent horizontal stratification (see figure 5 below for
the Atlantic).
9
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 5
1.4
Sound speed
It is known empirically that sound speed c varies as a function of T, S and z. Also,
sound speed is approximately horizontally stratified.
Many empirical relations c = c(T , S , z ) have been developed. The simplest equation is
that of Medwin and contains 6 terms:
c = 1449.2 + 4.6 T − 0.055 T 2 + 0.00029 T 3 + (1.34 − 0.01T )( S − 35) + 0.017 z
The domain of applicability is:
0 < T < 35 oC
0 < S < 45 ppt
0 < z < 1000 m
and the standard error amounts to 0.2 m/s.
Mackenzie’s equation is more accurate (error 0.07 m/s) and contains 9 terms:
c = 1448.96 + 4.591T − 5.304 × 10 −2 T 2 + 2.374 × 10 −4 T 3
+ 1.340 ( S − 35) + 1.630 × 10 − 2 z + 1.675 × 10 −7 z 2
− 1.025 × 10 − 2 T ( S − 35) − 7.139 × 10 −13 Tz 3
The domain of applicability is now:
10
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
− 2 < T < 30 oC
25 < S < 40 ppt
0 < z < 8000 m
1.5
Sound speed measurements
The expendable (X) version of the bathythermograph (BT), the XBT, measures T as a
function of z, using a known fall rate. c is then determined using one of the empirical
relations assuming constant S. This assumption is not always valid (especially in
coastal areas). Then, a velocimeter, which measures c directly, is preferable. An
expendable version of the velocimeter is also available (XSV).
Also available is expensive but recoverable equipment (usually deployed from
oceanographic ships) containing several sensors for measuring T, S (or conductivity),
P, c, dissolved oxygen, etc. (e.g. the CTD device, figure 6).
Figure 6
1.6
Sound speed profiles
The following situation is often encountered in deep-ocean areas, where the profile
can be divided into a few layers, see figure 7.
11
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 7
T-profile
Mixed layer: well-mixed layer of
isothermal water
Thermocline: T decreases rapidly with z
Deep isothermal layer
c-profile
Sonic layer (‘duct’).
c increases with z due to the hydrostatic
pressure effect. c-profile is linear with
gradient 0.017 s-1
Negative gradient in c
c increases with z due to the hydrostatic
pressure effect. c-profile is linear with
gradient 0.017 s-1
The sound channel axis is the depth corresponding to the sound speed minimum
between the negative c-gradient of the thermocline and the positive gradient of the
deep isothermal layer. Around the sound channel axis a sound channel is formed
within which sound energy is more or less confined by refraction.
At low latitudes the sound channel axis is at approximately 1000 m depth. It decreases
with increasing latitude and approaches zero (i.e. the axis lies at the sea surface) in the
polar regions.
The critical depth is the depth below the sound channel axis at which c equals the near
surface maximum. The depth excess is the vertical distance between the critical depth
and the seafloor.
This simple model does not apply in the winter season and in shallow water.
Figure 8 gives c-profiles measured in the summer season in the Mediterranean Sea.
The mixed layer and thermocline are clearly present.
12
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 8
The depth dependence of c poses a problem for echo sounders, which assume a
constant c in the water column when calculating seafloor depth from the two-way
travel time of the signal. When the actual depth-integrated value of c differs from the
assumed value, a correction must be applied. These corrections can be on the order of
several percent of the true water depth.
1.7
Sea surface
The sea surface is a reflector and scatterer of sound. It is a perfect reflector when
perfectly smooth (due to the large difference in acoustic impedance between air and
water). Reflection loss is no longer zero when the sea surface becomes rough (due to
the influence of wind). Sea surface roughness is specified in terms of wave height,
which is usually expressed by a single parameter: significant wave height H1/3. This
was originally defined as the mean height of the highest one-third of the waves, which
was found to be close to wave height estimated visually by a trained observer. The
modern definition of H1/3 is expressed in terms of the wave spectrum. Based on the
Pierson-Moskowitz (P-M) spectrum, representing the saturated spectrum independent
of duration of the wind and the fetch (i.e. the distance over open water over which the
wind acts from the same direction). The P-M spectrum is only a function of wind
speed U (measured at a height of 19.5 m). Since it was derived from mainly open
ocean observation of waves, the P-M spectrum is not thought to be valid in shallow
water regions. Wave measurements in shallow water (coastal applications) show an
enhancement of the spectral peak (‘JONSWAP’ spectrum), see figures 9 and 10.
13
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 9
Figure 10
If σ2 is the variance of the wave field, derived by integrating the area under the
spectral curve, then
H 1 / 3 = 4σ
14
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
σ is the corresponding standard deviation of the sea surface (the root-mean-square
(rms) wave height).
The wavelength, phase speed and period of the dominant sea surface waves (the
waves at the peak of the spectrum) are denoted λ, cph and Tp, respectively. The phase
speed is given by
c ph =
λ
Tp
.
The relation between frequency fp (= 1/Tp) and wavelength is given by the dispersion
relation for gravity surface waves (gravity being the primary restoring force):
(2πf p ) 2 =
2πg
 2πH 
tanh
.
λ
 λ 
with g the acceleration due to gravity.
For a water depth H = 100 m, Tp is plotted versus λ in figure 11 below.
Figure 11
For H = 100 m the sea surface parameters discussed are given as a function of sea
state (or wind speed) in the table below.
15
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Sea state
1
1, 2
2, 3
3, 4
4, 5
5
6
6
U [m/s]
1.6
3.3
5.4
7.9
10.7
13.8
17.2
20.7
cph [m/s]
1.9
3.8
6.1
9.1
12.2
15.8
19.3
22.5
Tp [s]
1.2
2.4
3.9
5.8
7.8
10.1
12.5
15.1
Fp [Hz]
0.83
0.42
0.26
0.17
0.13
0.099
0.080
0.066
λ [m]
2.3
9.0
23.8
52.5
95.0
159
241
339
H1/3 [m]
0.1
0.3
0.7
1.5
2.8
4.7
7.3
10.6
The breaking of waves produces subsurface air bubbles. Free air bubbles in the sea
are quite small as larger bubbles quickly rise to the surface. The bubbles only form a
very small volumetric percentage (the so-called void fraction β) of the water. Because
of the very large differences in density ρ and compressibility κ between water and air,
the suspended bubbles have a profound effect on sound propagation. A minute
amount of air ( β ≈ 10 −4 ) substantially reduces the speed of sound in the bubbly fluid.
The effect can be described by simple mixture theory:
c mixture =
1
[βρ air + (1 − β ) ρ water ][βκ air + (1 − β )κ water ]
with
ρ air << ρ water
κ air >> κ water
For β << 1 this can be approximated by
1
c mixture = c water

κ air 
1 + β
κ water 

with
1
c water =
ρ water κ water
Similarly
cair =
1
ρ airκ air
16
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
A volume fraction of air of only 0.01 % ( β = 10 −4 ) reduces the sound speed by a
factor of two (see figure 12). Notice that the sound speed of the mixture drops below
the sound speed of air (340 m/s) when β > 10 −3 !
Figure 12
In figures 13 and 14 we show results of a ‘sophisticated’ subsurface bubble model for
wind speeds of 5.4 and 13.8 m/s, respectively.
Figure 13
17
SEAFLOOR MAPPING – MODELLING - OCEANOGRAPHY
Figure 14
Aside from reflection loss, there are other acoustic effects associated with the
interaction of sound with the sea surface:
- a moving sea surface produces frequency-smearing and shifting effects (Doppler
effect);
- large and rapid fluctuations in amplitude;
- Lloyd mirror effect producing a pattern of constructive and destructive
interference between direct and surface-reflected signals.
18