From Pressure to Depth
Estimation of underwater vertical position
Havbunnskartlegging
og Inspeksjon
6.-8. Februar 2008
Geilo
Ove Kent Hagen
Avd Maritime Systemer
FFI
Underwater pressure measurement
Sea surface
Atmospheric
pressure
Water level
MSL
Pressure field
=
Hydrostatic pressure field
+
Dynamic pressure field
Pressure sensor
Dynamic near field:
• Current-Hull effects
Vehicle reference point
• Wave-Body interactions
Hydrostatic pressure
∂p
= ρg
∂z
ere
h
p
os
m
t
A
ean
c
O
Jorden
Earth
Jorden
z
Vehicle
• The pressure p equals the weight per unit area of the water and
atmosphere column above the vehicle
• There exists a 1-1 relationship between pressure and depth z
• Rule of thumb: 10 m water depth = 1 atmosphere
• Challenges:
– The density ρ depends on pressure and hence on depth
– Gravitational acceleration g depends on the vehicle’s
position
Density
Density of sea water
ρ0
• Depends on pressure
ρ > ρ0
ρ < ρ0
• Depends on temperature
Sa
lt
• Depends on salinity
ρ > ρ0
”Measuring” the density of sea water
• CTD (Conductivity, Temperature, Density)
– Pressure, p
– Temperature, T
– Conductivity, C
C

S = SPSS-78  , T , p 
 C0

• Salinity is estimated by UNESCO formula
”Practical Salinity Scale (1978)”
(PSS-78)
IES-80 density at atmospheric pressure
30
10 0
2
1030
• Density is estimated by UNESCO formula
25
1025
”International Equation of Sate of sea water (1980)”
1020
15
1015
1010
10 30
10 28
10 26
10 24
10 22
10 20
10 18
10 16
10 14
10 12
10 10
10 08
5
10 06
10
10 04
1005
10 32
ρ = ρ IES-80 ( S , T , p)
Temperature [degC]
(IES-80)
20
0
5
10
15
20
25
Salinity [psu]
30
35
40
1000
Hydrostatic pressure to depth from a
CTD profile
• Measure the conductivity C(p) and temperature profile T(p) in the water column
• Estimate the salinity profile S ( p ) = SPSS-78 (C ( p ), T ( p), p )
• Integrate the hydrostatic equation from vehicle depth to the water level
z
p
1
∫0 g(φ , Λ, z )dz = ∫0 ρ ( p) dp
Latitude and longitude
p
1
 1

z 1 + γ z z  g 0 (φ ) = ∫
dp
ρ IES ( S ( p), T ( p), p)
 2

0
A crude model of gravitation
CTD profile
UNESCO Pressure to Depth
Standard ocean: S=35 psu and T=0 °C
– Specific volume
V = VIES-80 (S , T , p) =
1
ρIES-80 (S , T , p)
– Specific volume anomaly δ = δ IES-80 (S , T , p) = VIES-80 (S , T , p) − VIES-80 (35,0, p)
p
p
1
1
z=
VIES-80 (35, 0, p )dp +
δ IES-80 ( S ( p ), T ( p ), p )dp
∫
∫
g(φ , p) 0
9.8 0
Standard ocean UNESCO equation:
- Integral: 4’th order polynomial fit in p
- Gravitation:
Geopotential height anomaly
- Cumulative numerical integration of the profile
- Thereafter, table look-up with linear interpolation
g(φ , p ) = g 0 (φ )(1 + γ p p)
International equation of gravity at surface
Increasing linearly with pressure (depth)
Hydrostatic pressure to depth below MSL
1. Subtract atmospheric pressure at
sea surface
CTD profile & Geopotential height anomaly
2. Use the standard ocean
UNESCO equation for pressure
to depth below the sea surface
3. Estimate geopotential height
anomaly from the CTD profile,
and add to depth
4. Subtract estimated water level
above MSL
Breiangen, December 2001
Slowly varying error
Surface wave induced pressure field
Predicted depth error due to dynamic wave pressure field
0
0.6
5
0.4
10
• Waves attenuate with depth
– High attenuation: wind waves
– Low attenuation: swells
15
0.2
z [m]
20
0
25
• The field becomes more regular with depth
Significant wave height: 5 m, Peak time period: 8 s, Water depth: 80 m
30
5
-0.2
JONSWAP
JONSWAP
JONSWAP
JONSWAP
35
4.5
-0.4
40
surface wave spectrum
at 5 m depth
at 10 m depth
at 15 m depth
4
45
-0.6
3.5
-150
-100
-50
0
x [m]
50
100
150
200
• Swell and wind waves:
– Period: 0.2 – 15 s
– Frequency: 5 – 0.06 Hz
3
S(ω) [m2s]
50
-200
2.5
2
1.5
• No longer 1-1 between pressure and depth
1
0.5
Fast varying error
0
0
0.05
0.1
0.15
Frequency [Hz]
0.2
0.25
0.3
Near field effects
• The pressure measurement depends on the vehicle’s water referenced
velocity and the sensor’s location on the hull:
• Counteract through design
• Compensate through model
• Wave-body interaction:
– Long wave approximation:
• Wave length >> vehicle dimension
• Vehicle (neutrally buoyant) follows the particle path in the waves
– Otherwise:
• Scattering potential caused by the vehicle’s presence in the
incoming waves
• Radiation potential caused by the vehicle’s response to the
incoming waves
– The motion may be counteracted by the vehicle’s control system
Uncertain fast and slowly varying errors
Robustness needed
Precise depth estimation using NavLab
• Combine UNESCO pressure to depth with inertial
CTD
Pressure
IMU
GPS
DVL
Tide
navigation
• Inertial navigation estimates the vehicle’s short term
motion with high precision – filters wave induced
“pressure sensor noise”
Atm
Robust
Robust
noise
noise
parameters
parameters
Optional
Optional
Unesco
Unesco
P
P
R
R
E
E
P
P
R
R
O
O
C
C
E
E
S
S
T
T
I
I
M
M
A
A
T
T
O
O
R
R
Cmp
NavLab OneClick
Automatic processing controller
S
S
M
M
O
O
O
O
T
T
H
H
I
I
N
N
G
G
E
E
X
X
P
P
O
O
R
R
T
T
Smoothed
Position
Attitude
Depth
Pressure
Test with HUGIN 1000
La Spezia, Italy:
• Low amplitude swell
• Shallow water
• Flat seafloor
Inertial Measurement Unit: iXSea IMU 120
Doppler Velocity Log: RDI WHN 600 kHz
Pressure sensor: FSI Mirco CTD
HUGIN 1000 was operated from R/V Leonardo of the
Multi beam echo sounder: EM 3000
NATO Undersea Research Centre
NavLab post-processing: smoothed depth
depthm error (bias and total) and KF-model (1 and 3 sigma)
0.2
std =0.10262
• Bias oscillation period ~ 7.5 s
0.15
• Sea floor depth ~ 17 m
0.1
0.05
• HUGIN’s depth ~ 6 m
[m]
0
-0.05
• Wave length of the swells
causing the oscillations ~ 100 m
-0.1
-0.15
-0.2
-Depth [m]
5110
5120
5130
5140
5150
Time [s]
5160
5170
5180
-5.6
-5.8
-6
-6.2
-6.4
5110
5120
5130
5140
5150
Time [s]
5160
5170
• The long wave approximation is
valid – HUGIN follows the wave
motion
5180
Altitude control in long waves
•
•
•
Waves change the vehicle’s altitude while the pressure stays the same
The control system counteracts this by going deeper/shallower
The pressure increases/decreases – altitude decreases/increases
Same pressure
-Depth [m]
Altitude increase
Pressure increase
Altitude decrease
-5.6
-5.8
-6
-6.2
-6.4
5110
5120
5130
5140
5150
Time [s]
5160
5170
5180
EM 3000
bathymetry
Only hydrostatic
pressure to depth
conversions
Uses the output
of Preproc in
NavLab
EM 3000
bathymetry
Kalman filtered
depth
This is
achievable in
real-time
Uses the
output of the
Estimator in
NavLab
EM 3000
bathymetry
Filtered by optimal
smoothing
This is achievable
in post-processing
Uses the output of
Smoothing in
NavLab
Conclusion
•
By combining inertial navigation with the UNESCO pressure to depth
conversions, precise depth estimates can be made for underwater
vehicles, even when operating in the surface wave pressure field
•
Applications for improved depth estimates:
– Improve post-processing of digital terrain models, and seabed
imaging
– Improve real-time depth control of underwater vehicles
– Improve bathymetric measurement inputs to terrain navigation
•
References:
–
–
–
–
–
Fofonoff & Millard: ”Algorithms for computation of fundamental properties of seawater”,
UNESCO Technical Papers in marine science 44, 1983
Hagen & Jalving: ”Converting Pressure to Depth for Underwater Vehicles”, FFI-Rapport,
(TBP)
Willumsen, Hagen, and Boge: ”Filtering depth measurements in underwater vehicles for
improved seabed imaging”, Oceans Europe 2007, Aberdeen
www.navlab.net
www.ffi.no/hugin