THE ATLAS ALPHA DOPPLER NAVIGATOR SYSTEM
by W .
S te d tn itz
(*) and H.
H e lm s (* * )
Atlas E lektronik, Bremen
This paper was prepa red fo r the D e u ts c h e Gesellschaft fiir O rtu n g
und Navigation and is a slightly m o r e elaborate fo r m of the a u th ors’
lecture on the same s u b je c t given at the 10th International H y d rog ra p h ic
C on feren ce in A p r il 1973.
PRINCIPLE
Rapid progress in the fields o f electronic and d igital com puter
technology has proved a strong stimulant to doppler navigation. A pre­
requisite fo r the application o f the doppler technique is the know ledge
and m astery o f numerous physical param eters both in w a ter — the
propagation medium — and onboard the vehicle itself, m ore particu larly
the param eters required fo r the correct setting o f the gyroscope. This
subject w ill be dealt w ith in the second part o f this paper. The ability
to interpret both in mathem atical and statistical term s the processes
involved in doppler measurement is essential if the accuracy o f the
course made good by a vessel em ploying doppler is to be predicted.
The A lpha doppler system is based on the acoustic doppler effect,
and p rim arily measures the ship’ s speed relative to the gro u n d , fo r speeds
ranging between — 5 and 36 knots, w ith a resolution o f 0.01 knot
(5 m m /sec). Th e track can then be calculated on the basis o f this speed.
Determ ination o f the track is independent o f the follo w in g factors :
— w ater tem perature
'
salinity
W h e n using
degree o f water pollution
► special A lph a
w ater depths w ithin the range 0.5 to 600 m
D oppler
settings or adjustm ents o f equipm ent
speed up to 36 knots
W h en using double
beam arrangem ent
(*) A u t h o r o f first tw o c h a p te rs : P r in c ip le , a n d A p p lic a tio n s .
(**) A u t h o r o f c h a p te r 3 : Test R e s u lts a n d A ccuracy o f P o s it io n in g .
—
—
—
—
—
other ships w ith in the area o f operation
land stations
currents
w ind drift
environm ental conditions (e.g. mountains,
ships, or large buildings, affecting radio
navigation)
— ground structure, but with certain restric­
tions
W h en using any
doppler technique
Speed is determ ined by assessing the difference in frequency between
the transm itted and the received signal. In doppler measurement travel
tim e does not have to be assessed, the doppler frequency being independent
o f water depth. A doppler frequency shift occurs during transition from
one medium to another if there is relative movement between the two.
Thus, doppler frequ en cy shifts m ay occur :
1. D uring transition from the transducer to the surrounding w ater;
2. In the water, if there are layers with different sound velocities;
3. During reflection at the bottom, if there is a current over the
bottom.
These shifts taken together total the value that the Sonar Doppler
measures. As long as bottom contact is possible, this value w ill in all
cases be proportional to the speed made good over the ground.
T h e intensity o f reflection is not important. Fluctuations can be
compensated to a large extent by AGC am plifiers and filters. Measurement
above a m uddy bottom is, therefore, as feasible and as accurate as measure­
ment above rocky ground.
T hree factors influence the measurement of speed, these being the
acoustic conditions, the finite travel tim e of sound, and the influence o f
sound velocity.
1. Acoustic conditions.
Undisturbed acoustic conditions fo r transmission and reception are a
prerequisite for an accurate Doppler speed and track measurement. The
flat shaped Alpha D oppler transducer lends itself to an ideal flush mounting
in the ship’s hull. Nevertheless the installation position must be carefully
selected in fast ships in order to prevent acoustic disturbance from air
bubbles. W h ere speeds exceed 30 knots, the hull o f the vessel forw ard
o f the transducer array must meet certain hydrodynam ic requirem ents
as cavitation must not be allowed to occur in this area.
2. Finite travel time of sound.
Sound takes a certain tim e to travel from the ship to the sea bottom
and back. I f the ship is m oving at very high speed it w ill have covered
a certain distance in the tim e between transmission and reception.
H owever, there is on ly a m inor shift in the radiation angle (below 1" at
50 knots) which is com pletely compensated by a double-beam arrangem ent.
3. Influence of sound velocity.
W o rk s o f reference usually give the doppler form ula expressing the
relation between doppler frequency and ship’ s speed as :
2V
A / = f s — cos a
(1 )
A f being the doppler frequency, f„ the transm itter frequency, V the ship’s
speed, c the sound velocity (dependent on temperature and salin ity), and
a the radiation angle. c//s = X. = acoustic wavelength in water.
F orm u la (1) shows, inter alia, that doppler frequency is not on ly a
function o f the ship’s speed, V, but also o f the transm itter frequency,
sound velocity and radiation angle. H ow ever what is required is m erely
the proportional relationship to the ship’ s speed.
The Alph a D oppler System, too, w ork s on Ibis principle. H owever,
in this system due to the special geom etry of the transducer the radiation
angle a is itself a function o f sound velo city (see figs. 1 and 2 ) :
X
cos a = —
3a
a being the spacing between transducer elements.
( 2)
Substituting this last
expression in ( 1 ) :
2V
A / = —
(3)
3a
The doppler frequency is now on ly a function of the variable speed o f
the ship and a constant mechanical qu an tity a.
X.Y
F ig. 1. — Schematic arrangem ent o f a b o t t o m m ou n ted transducer array sh o w in g
the cable leads.
3a -------------------------------- H
F ig. 2. — Three ad jacen t tr an sd ucer elements show in g geom etry
o f tr a n sm iss ion wave length X.. Radiation angle a.
\ = 3a cos a
Th ere are two pairs o f transducer arrays, each pair set up in the tw o
principal axes o f the ship. Each a rra y (see fig. 1) consists o f 18 elements,
and the fact that there are m ore transducer elements in one direction
then in the other, results in tw o different beam widths, i.e. ± 3° and
± 8 ” . In developing the transducer array a com prom ise has been made
between the amount o f transducer m aterial and the fluctuation o f the
doppler values measured. The statistical features o f the doppler signal
are characteristically dependent on the w idth o f the sound beams. The
second part o f this paper describes in detail how these features are reflected
in the measurement results.
Compounds dissolved in the w ater affect sound velocity, but their
influence on the doppler frequen cy can be compensated by automatic
changes in the beam depression angle.
P articles floating in the w ater reflect sound, and this gives the
erroneous impression that it is a doppler frequency induced by flow in g
water. H owever, the sound reflected by the water reaches the receiving
transducer somewhat earlier than the sound reflected by the ground. T im e
separation is how ever feasible by adopting the pulse mode o f operation.
A tim e filter, which adjusts autom atically according to depth, separates
the w ater echoes from the ground returns. If required it is therefore
also possible to measure relative speed in relation to the water.
It has been proved statistically that fluctuations o f the doppler signal
in the pairs o f beams are correlated. By means of special circu itry in
the frequen cy evaluation circuit it is possible to obtain a large measure
o f com pensation for equipm ent d rift. During pulse operation such d rift
is less than 5 metres per hour.
In this connection attention should be drawn to an effect leading to
disturbance. Sea-water is saturated with gas, in particular after a storm.
W h en a ship is underway the w ater im m ediately around the hull is
warm ed by the ship, particularly in cold weather. W ith increasing
warmth, the gas solubility decreases and the water begins to effervesce.
This may induce loss o f the sonar contact, and the d rift rate w ill increase
abruptly. Sim ilar phenomena can sometimes be observed in calm waters
o f harbour basins and, at times, with the ship at anchor when methane
gas from sewage rises from the bottom.
Loss of sonar signal due to ship-induced bubbles never occurs in
the case of a stationary ship, at anchor or moored in flow in g water, or
if the vessel is m oving very slow ly in calm water. The very low relative
speeds are sufficient to prevent the form ation o f air bubbles.
It is obvious that turbidity due to the ship’ s m ovem ent astern can
also be a cause o f interrupted sonar contact on account o f the form ation
o f a substantial amount o f air bubbles. Precision measurements are in
this case always subject to disturbance.
Doppler navigation is a dead reckoning method, the ship’s position
being calculated at any tim e from exact measurements of track length and
direction since departure from a known starting point. T h e track length
is derived from the measurement o f speed relative to the ground.
The basic track length is the resultant of the track made good during
a given tim e interval at speed v,: in the ship’s longitudinal axis and at
speed vy in the transversal axis. The overall track and the corresponding
position o f the ship, P, can be computed by summing these two track
elements vectorially.
As is w ell known, doppler measurement also takes account o f current
and wind influences.
The track made good through the water, ASW (see fig. 3) in a given
tim e interval At is the vectorial sum o f the ship’s steered track ASK, and
the wind set A W . W h en the d rift due to current AT, is added to AS„
vectorially, the track made good over the ground AS<4, is fin ally obtained.
The track made good can also be divided into tw o ship-referenced
components by means o f the true course indicated by the gyroscope, or
supplied by a computer. These two components are the one in the ship’ s
longitudinal axis, AX<j, and the other in its transversal axis, A Y C. Both
are computed from the speeds measured by the doppler system.
In cases where w ater depths exceed 600 m and consequently no
echo contact w ith the ground is possible, measurements can on ly be carried
out by reflection against volum e reverberation from the water. In this
case the doppler system measures the relative tracks m ade good w ith
respect to the water, AX,V and AY,V, and from these the resultant track
element ASW can be derived. This allows wind drift to be taken into
account, but not d rift due to currents.
The Alpha Doppler System uses fou r independent sound beams for
the doppler measurements, with a fifth beam for depth measurements.
T hree frequency bands are used for separation — a band each fo r doppler
I
/
/
ASq
T ra c k m ade g o o d over the g ro u n d
ASw
T ra c k m ade g o o d th ro u g h th e w a te r
AT
D r if t (cu rre n t)
AW
W ind set
AS^
S hip's steered tra c k
AXq
L o n g itu d in a l tra c k m ade g o o d measured b y d o p p le r
AYq
Transversal tra c k m ade good m easured b y d o p p le r
AX yy
R elative lo n g itu d in a l tra c k m easured by d o p p le r against w a te r reverberations
A Y yj
R elative transversal tra c k m easured against w a te r reve rb e ra tio n s
Fie*, .‘i- — C om position o f the course vectors.
measurement in the ship’s two principal axes and a third fo r independent
depth measurement.
The transm issions are tim e-locked in order to m aintain separation
o f the in form ation in each pair o f beams. A distinction is made in terms
o f tim e between :
M ain pulse
M ain pulse
R everberatin g echo
Ground return
R everberatin g echo
Ground return
ahead
astern
ahead
ahead
astern
astern
—
—
—
—
—
—
starboard
port
starboard
starboard
port
port
The pulse tim e pattern m ay be changed, depending on whether the
reverberating echo or the ground return is used, in order to obtain optim um
measuring conditions fo r each case.
T o this effect both manual and autom atic selection o f the operating
mode have been incorporated. The duration o f the main pulse can be
adjusted according to the depth. The product o f pulse length and repeti­
tion frequ en cy is, how ever, maintained constant.
F or cost reasons the transm itter circu itry has been sim plified so
that it is on ly possible to transm it one beam at a tim e in each main axis.
H owever, the two doppler measuring beams and the depth beam can be
received simultaneously. In this case, the channels fo r the doppler beams
are separated according to direction and time, and the depth beam
according to direction, tim e and frequency.
E ith er o f two modes o f operation m ay be selected :
G r o u n d track (the norm a) m ode).
T w o ultrasonic beams, one pointing forw ard and the other astern, are
transm itted to the ground, one shortly after the other. As the ship
moves, the ground-reflected sound is received with a frequ en cy shifted
b y the doppler frequency. The generator is clock-controlled so that the
main pulse stops as soon as the return echo is received.
W a t e r track.
In this mode o f operation the ship’ s speed is measured relative to
the water. A special w in d ow control fo r tim e ensures that on ly the
echoes fro m the w ater mass (volum e reverberation) are used fo r doppler
measurement, w hatever the depth.
This mode o f operation is o f particu lar value in the fo llo w in g cases :
— Th e depth o f the w ater is such that reception o f the ground returns
is impossible.
— W h en the ship’ s engines are to be checked or adjusted, the speed
is then required to be measured in relation to the water, and a ground
return is deliberately avoided.
A P P L IC A T IO N S
Th e A tlas A lpha D oppler System has been developed as a modular
system ; fou r o f the modules being as follow s :
— The Atlas Alpha Doppler Log measures a ship’ s speed in the longi­
tudinal axis only. The track made good is determ ined by a single integra­
tion o f speed. The Atlas Survey System Susy m ay be connected to this
log system to obtain m ore accurate readings.
— The Atlas Alpha Doppler System fo r satellite n avigation measures
the longitudinal and transversal components o f a vessel’s speed. The
m ccia u iu u
vrtiu^a
ui c
iu u
im u
iiit
^ixij_r s
csdLUiiii^ n o v i ^ a i i u i i
oiv. 111,
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processed, and provide m ore accurate positioning, the ship’ s speed being
an im portant input when m aking satellite fixes.
— The Atlas Alpha Doppler Docking- System is used fo r m on itorin g the
v ery lowest speeds or sm allest movem ents o f a vessel, in particular during
docking operations. F or this purpose three speed com ponents are mea­
sured — the longitudinal speed, the transversal speed at the bow, and
the transversal speed at the stern. The last digit o f the indicator reads
in hundredths o f a knot.
— The Atlas Alpha Doppler Navigator is the most com prehensive module
in the A lph a D oppler System, being able to accomplish all the navigational
functions o f a vessel. It measures both components o f the ship’s speed.
A ll other navigational data are determ ined by the computer, utilizing
compass inform ation, position and correction data, as w ell as constants.
F o r this purpose the com puter carries out the follo w in g functions :
— In terrogation of the counters in order to store the longitudinal and
transversal track elements fo r one position-com puting cycle.
— Computation o f the d rift angle from the longitudinal and trans­
versal speed values. Output o f this angle to an indicator and, optionally,
to a tape puncher or teleprinter.
Interrogation o f the gyro to obtain the steered track angle fo r one
position-com puting cycle.
— Computation o f the track made good over the bottom, and output
o f the distance run to an indicator, tape puncher or teleprinter.
— Computation o f the gyro error from ship’s speed, gyro track angle
and the latitude o f the ship’s position.
— Computation o f the true track angle from the gyro steered track
angle, gyro error and ship’ s d rift. Output o f the true track angle to an
indicator, tape puncher or teleprinter.
— Computation o f the elements o f the track made good in the NorthSouth/East-W est geodetic reference system from the elem ents o f the
ship’ s track made good in the longitudinal and transversal directions and
from the true track angle.
— Integration o f the elem ents o f the track made good into the geodetic
referen ce system.
— Computation o f the ship’s actual position in latitude and longitude
taking account o f the reference ellipsoid and the in itial coordinates.
Output to an indicator, tape puncher, or teleprinter.
— Computation on a plotter o f the rectangular coordinates fo r the
ship’s actual position in the projection used, taking into account scale,
the reference coordinates fo r the plotter and those fo r the projection.
Output o f these coordinates to the plotter.
— Interrogation o f the digital clock and output o f the time.
A ll the referen ce and correction values, as w ell as all constants — the
scale factor o f the chart, the ship’s initial position at the start o f a voyage,
and com parative values from other navigational aids, fo r exam ple — neces­
sary for these com putations may be fed into the com puter via the tele­
printer.
Most o f the in form ation required fo r the solution o f the underm en­
tioned problems is th erefore available in this system.
— Storage o f target coordinates. Computation o f the most suitable
course to the destination point, and the autom atic holding o f this course
with an autom atic pilot. Continuous m onitoring o f the course and, if
necessary, its correction.
— Integration o f the doppler navigation system w ith other navigation
systems, fo r exam ple w ith H i-Fix, or satellite or inertial n avigation systems,
in order to increase accuracy, by using the data obtained fro m these
systems to the best advantage.
— Prevention o f collisions, by determ ining the courses and speeds
o f vessels in the vicin ity, and thus forecasting the tim e and distance o f
the closest point o f approach CCPA).
— Coordination of position and depth values in the case o f surveys
whose object is com putation o f depth contours (tw o-dim ensional surveying).
TEST R ESULTS A N D A C C U R A C Y OF P O S IT IO N IN G
This part o f the article is an extract from a com prehensive report
on tests o f the A tlas Alpha D oppler N a vigator system fo r accuracy o f
positioning. Th e general experience gained from the tests carried out
between Novem ber 1971 and March 1972 w ill be described. T h e problem
o f navigational accuracy w ill also be considered, and the results w ill be
given.
A m ajor problem in testing a navigational system is the determ ination
o f the accuracy o f position finding under the prevailin g environm ental
conditions. This is a problem that obviou sly cannot be solved in the
laboratory; the system has to be operated in the field. T his requires a
considerable amount o f time and m oney because w ith the doppler principle
— a dead reckoning technique — numerous measurements have to be
carried out often over long distances and w ith the com plete system
operating in tw o dimensions. T h is is the on ly w ay in which one can
be assured o f acquiring valid com parative statistics for positional errors.
In this connection uncertainty in the determ ination o f the course b y gyro
compass — an uncertainty w hich is m ainly due to the vessel’s accelera­
tion — itself proves to be a great problem.
T o provide a reference system for the doppler system tests a H i-F ix
chain was available, thus perm ittin g alm ost continuous com parative deter­
m inations o f position.
Another problem durin g accuracy
various environm ental conditions each o f
the doppler navigation system. Figu re
gives in form ation regarding the effect of
fo r their elim ination.
measurements results fro m the
which can lead to errors affecting
4 lists these influences and also
these errors and the possibilities
It is im portant to realize that only a part o f the statistical speed and
course error is due to external effects, what remains o f these tw o errors
form s the true positional error o f the system. A ll the other errors can
be assessed and elim inated by either physical measurements or mathem atic
com putation. The tests took into account most o f the existin g environ ­
m ental conditions.
O n ly those test data o f general interest w ill be given.
the scope o f the tests and the efficien cy o f the system.
T est period :
T est area :
T est vessel :
T h ey illustrate
N ovem ber 1971 to March 1972
W estern Baltic Sea, Skagerak
Hans Bürkner
E 71, E ckernforde
Number o f runs
Test runs to be evaluated
Of
Of
Of
Of
Of
Of
less than 20 n.m.
20-40 n.m.
40-60 n.m.
60-80 n.m.
80-110 n.m
more than 110 n.m.
5
7
10
9
3
1
T
M axim um depths (m easured
Skagerak) :
E nvironm ental conditions :
in
otal
:
35
2000 n.m.
approx.
4000 n.m. approx.
> 700 hours
— 5 . . . . 23 knots
Translation : 0 . . . . 6.5 . 10-3 g
Rotation :
0 . . . . 1.5 . 10 ~2 g
T o ta l distance o f test runs :
T o ta l distance covered :
T otal duration o f operations :
Speed range :
E ffective acceleration o f vessel :
the
500 ____ 600 m
(depending on bottom topography)
W in d velocity : 0 . . . . 11 knots
Sea state :
0. . . . 7
(w ithout system fa ilu re)
Currents :
0 . . . . 2.5 knots
Tem peratures : — 1 . . . . + 14 °C
I n flu e n c e fa c to rs
(m e a s u re m e n t
p a ra m e te rs )
In s ta lla tio n
in flu e n c e s
R a nge o f m easure­
m e n ts d u r in g
te s t p e r io d
In s ta lla tio n
Speed
-
5 . . . + 23 kt
E f f e c t o f e rro rs
E r r o r o f re fe re n c e
C o n s ta n t c o u rs e e rro r
and th e s pee d fa c to r
E rro rs can be m easured
an d e lim in a te d b y
c o m p u te r
In c lin a tio n s are d e p e n d e n t on
s h ip t y p e a n d speed e rro r
e lim in a te d b y c o m p u te r
B u b b le e ffe c ts in te r fe r e w ith
D o p p le r s p e c tru m ; s ta tis tic a l e r ro r
Vessel s
m o tio n
E ffe c tiv e
a c c e le ra tio n
T ra n s la tio n :
0 . . . 6 .5 . 1 0 - 3 g
R o ta tio n :
0 . . . 1 .5 . 1 0 - 2 g
W in d
Sea
C u rre n t
T e m p e ra tu re
S a lin ity
0 . . . 11 k t
0...7
In c lin a tio n o f vessel and
spee d e rro rs
Vessel's in c lin a tio n
P a rtly e lim in a te d b y c o m ­
p u te r o r e le c tr o n ic u n it
R e sidual s ta tis tic a l e rro r
Vessel's in c lin a tio n
e lim in a te d b y c o m p u te r
C ourse e r r o r d e p e n d e n t o n
a c c e le ra tio n in te n s ity .
D u ra tio n a n d re p e titio n
d e p e n d e n t o n g y ro ty p e
P a rtly re s id u a l s ta tis tic a l
c o u rs e e rro r
S ta tic a n d d y n a m ic vessel
in c lin a tio n s ; speed e rro r
e lim in a te d b y c o m p u te r
P a rtly e lim in a te d b y
c o m p u te r.
C om pass o s c illa tio n s , g im b a l e rro r
Vessel in c lin a tio n
C hange o f s o u n d v e lo c ity and
D o p p le r fre q u e n c y ; speed e rro r
E lim in a te d b y a u to m a tic
r a d ia tio n angle c o r re c tio n ;
Vessel's in c lin a tio n
0 , . . 2. 5 k t
-
1° . . . + 1 4 °C
Pressure
T h e re fo re a - D o p p le r
A ir
In te rfe re n c e o f D o p p le r s p e c tru m ,
E r r o r e lim in a te d p a r tly
Im p u ritie s
p o s s ib ly e c h o fa ilu r e ;
b y c o m p u te r a n d p a r t ly
b y e le c tro n ic u n it. R e sidual
s ta tis tic a l speed e r ro r
s ta tis tic a l s pee d e rro r
E n v iro n ­
m e n ta l
in flu e n c e s
E r r o r e lim in a t io n
D e p th
C o n tin u o u s wave
Pulse o p e ra tio n 1 >
Pulse o p e ra tio n 11 )
G ro u n d and
w a te r re tu rn
G ro u n d r e tu r n d is tu rb a n c e
by re v e rb e ra tio n
4 . . . 600 m
W eak e c h o a m p litu d e s , a m p l.
d is tu rb e d b y re v e rb e ra tio n ,
c u rre n t e rro r ;
s ta tis tic a l s pee d e rro r
E r r o r e lim in a te d p a r tly
b y c o m p u te r a n d p a r tly
b y e le c tro n ic u n it.
R e sidual s ta tis tic a l
speed e r ro r
D e p th change per
u n it o f tim e
M ay in flu e n c e D o p p le r s p e c tru m
(s h o u ld n o t n o r m a lly o c c u r),
e c h o d is tu r b e d b y d e p th
re a d ju s tm e n t ;
s ta tis tic a l s p e e d e rro r
E rro r e lim in a te d p a r tly
b y c o m p u te r, p a r tly b y
e le c tr o n ic u n it .
R e sidual s ta tis tic a l
speed e rro r
N a tu re o f th e
g ro u n d
D is tu rb a n c e o f th e D o p p le r
s p e c tru m b y in h o m o g e n e ttie s
(m a te ria l & s tru c tu r e ) ;
s ta tis tic a l s pee d e rro r
E r r o r e lim in a te d p a r tly
b y c o m p u te r , p a r tly b y
e le c tr o n ic u n it.
R e sidual s ta tis tic a l
speed e rro r
F ig . 4. — External influences e ffectin g the a c cu racy o f po s itio n d eterm ina tion
by sonar doppler.
Table of measurem ent parameters to he taken into consideration.
Nature o f seabed :
Sandy (in western B a ltic)
R ock y (in the Skagerak)
D ifferen t methods w ere used fo r fixin g reference positions as follo w s :
(a ) bearings o f the fo llo w in g landmarks - Eckernfôrde pier, beacons
m arkin g the L an gh ôft tw o mile measured distance, and K iel L ig h t­
house;
(b ) bearings of seamarks including various buoys and Feh m arn light
vessel;
( c ) using the E ck ern fôrd er Bucht-Fehrm an-Olpenitz H i-F ix chain;
( d ) using dead reckoning w hen speed and course w ere constant.
D uring the test phase the doppler speed sensors operated w ithout any
h ardw are failure. Fortunately, no echo disturbances, such as air bubbles,
were experienced up to the m axim um test speed o f 23 knots. However,
during the vessel’ s m ovem ents astern there w ere echo failures o f fa irly
long duration due to air bubbles. These resulted in measurable positional
errors. A t present this problem can only be solved by additional updating
o f positions used for position correction. T o what extent air bubbles on
the ship’ s bottom can be prevented from com ing into contact w ith the
surface o f the transducer by hydrodynam ic measures rem ains to be
investigated.
B efore discussing the results o f these tests the possible errors
the doppler navigation system in general must b riefly be mentioned.
of
Basically, three types o f errors can be distinguished :
a) System atic static errors
b) System atic dynam ic errors
c ) Statistical errors.
Perm anent statistical errors are in fact the lim itin g factor fo r posi­
tional accuracy.
W h en possible the system atic static errors o f the speed sensors,
which are due to tolerances accepted by the transducers them selves and
their installation, should be determ ined by taking several measurements
w ith each system as soon as it is installed. These errors can then be
elim inated in the com puter program by corresponding constant correction
factors fo r the speeds.
T h e system atic dynam ic errors o f speed are caused by deviations
o f the transducers from the horizontal position as a result o f pitching or
ro llin g or the trim o f the vessel. Considerable reduction o f error is
achieved by using a double-beam configuration. Figu re 5 shows which
speed errors rem ain despite the use of this configuration. A t pitching
am plitudes o f < 3", for exam ple, a negative relative longitudinal track
error o f < 0.1 % w ill occur. F o r the m ajor part o f the trials this
condition was satisfied, i.e. these inclination errors were negligible. If the
doppler system is to operate accurately at even greater deviations from
the horizontal the vessel’ s inclinations w ill have to be measured, and the
speed measurements corrected accordingly
A m on g the statistical speed errors which cannot be determ ined by
measurements arc the doppler fluctuation errors caused by the finite beam
w idth fo r em ission and reception o f sonar energy — these errors have
distance
error of measured
F ig . 5. — In clina tion errors fo r the Alpha Doppler Sonar beam arrangement.
very short correlation periods — added to which there is the effect o f the
structure and nature o f the seabed, the depth variations and im purities
in the water, for which the correlation periods m ay be short or long.
T rack errors due to statistical speed errors show a certain smoothing
which increases with the distance covered. This im plies that absolute
statistical track errors do not increase in proportion to the distance covered,
but approxim ately in proportion to the square root of that distance.
These static, dynam ic
and statistical speed
errors, taken
together,
position errors whose m agnitude is dependent on the distance
covered, which is usually substantially greater in the longitudinal direction
than in the transversal direction. The effect o f the transverse speed
errors, although also depending on the distance run, is thus smaller, being
ap proxim ately proportional to the arc subtended by the angle o f the
vessel’s drift. This angle of d rift is given by the ratio between the
transversal and the longitudinal speeds (in this case the angle o f d rift
is the angle between the true course and the true track).
fo r m
The system atic static error o f the course sensor depends largely
upon the in stallation ’s tolerances. The m agnitude o f this error depends
on the deviation between the gyro compass azimuth reference and the
reference axis o f the longitudinal speed sensor. This error should also
be determ ined, if possible, by m aking several test runs with each system
after installation. In order to obiain accurate position calculations it
w ill then be necessary to feed the correction to be applied to the course
sensor into the computer.
A com m ercial gyro compas (Anschutz Standard IV type) w ith liquid
dam ping is utilized as course sensor in the doppler system. T h e values
transm itted from the gyroscopic compass to the com puter have an
accuracy o f better than 0.1” . The static compass error amounts to
only ^ 0.1 °.
Since it is possible to calculate the so-called speed error w ith sufficient
accuracy, it is the dynam ic errors caused by the vessel’s acceleration which
constitute the m ajor problem in course determ ination.
L iq u id dam ping is provided for the gyroscopic sphere which is
suspended like a pendulum so that it m ay freely accede to the northerly
directional moment. However, accelerations of the vessel im part precession
moments causing the gyroscope sphere to oscillate.
T h e phenom enon for a vessel accelerating from 0 to 30 kt in 5 minutes
in a n orth erly direction has been described in technical literature elsewhere.
Figu re 6 is a teleprinter printout for a fictitious exam ple o f this
phenomenon.
T h e m axim um am plitude o f the resulting azimuth deviation amounts
to ap proxim ately 1.13“. Even w ith a constant northerly course there
would be a consequent error o f position. Although the angular error
dim inishes after some hours to zero, the total position error (resulting
from the integration of all the errors) rem ains o f considerable magnitude.
In practice horizontal accelerations in north-south and east-west directions
are gen erally a fa ir ly continual occurrence during m anoeuvring operations.
As a result there are coupled oscillations in the gyroscopic sphere that are
no longer controllable, and accordin gly errors in azimuth, and thus position,
are produced.
M ost acceleration oscillations can only be avoided by using other
methods o f suspension for the compass — fo r example gyroscopes
suspended by their centre of gravity. The horizontal stabilization and
orientation to north o f the gyroscope is then effected by accelerom eters
via electronic control-loops having special low-pass characteristics. There
already exist such compass systems entailing very small azimuth errors
2.4 9
4 .9 7
7 .4 6
9 .9 4
1 2 .4 3
14.91
1 7 .4 0
1 9.89
2 2.37
2 4 .8 6
2 7.34
2 9 .8 3
3 2 .3 2
3 4 .8 0
3 7.29
3 9.77
4 2 .2 6
4 4 .7 4
4 7 .2 3
4 9 .7 2
5 2 .2 0
5 4.6 9
5 7.17
5 9 .6 6
6 2 .1 5
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101.92
104.40
106.89
109.38
111.86
114.35
116.83
119.32
121.81
124.29
126.78
129.26
131.75
134.23
136.72
139.21
141.69
144.18
146.66
149.15
151.63
T im e
(m in )
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A z im u th
d e fle c tio n (°)
Fio. fi. — Sim u lation o f azim uth oscillation in the g yroscopic sphere o f an A n schütz
Mark IV g yro com pass fo r vessel acceleration f r o m 0 - 3 0 kt in 5 min
in a n ortherly direction .
even under extrem e acceleration conditions, but they are very expensive
and at present they can h ardly have com m ercial applications.
In order to reduce considerably the effects o f such azimuth errors on
position determ ination another m ethod was successfully tried. A com puter
program based on a m athem atical compass model was prepared fo r use
in the processing com puter used in the sonar doppler navigator. The
oscillations o f the gyroscopic sphere and accordingly its azimuth error
w ere thus calculated continuously during the tests by means o f this
program . This was also done fo r the vessel’s accelerations w hich can be
derived fro m the doppler measurements. The accuracy of this azimuth
error calculation naturally depends on the accuracy o f the m athem atic
gyroscopic m odel used. A lth ou gh only an incom plete mathem atical model
has hitherto been em ployed, the success obtained — in the form o f
considerable increase in positional accuracy — should encourage the
com pletion o f this m athem atic m odel. T h e advantage o f this m ethod
essentially lies in the fact that fo r a reduction o f the azimuth error by,
say, a factor o f 5 the position error depending on it is reduced in the same
proportion. Values o f this order were obtained during the tests. Th e
im portant fact is that the influence o f each acceleration disappears
asym ptotically in the course o f tim e both in the mathem atical model
calculations and by dam ping in the gyroscope itself.
The measurements carried out w ith the doppler system w ere exam ined
fo r short term fluctuations o f the sensor, fo r long term track errors,
and fo r overall errors. Short term fluctuations o f the speed sensor, for
example, are revealed by unsteadiness in the speed indications.
Uncorrected sensor data, recorded every 1.6 sec on the punched tape
during some o f the test runs were used for this investigation. The resolu­
tion o f the speed data amounted to 0.01 kt, and that of the course data
to 1.3'. It was decided to evaluate only those sections w ith course and
speed conditions as constant as possible, and speed and course variations
w ere thus largely avoided.
Figures 7, 8 and 9 show the results o f these statistical investigations
in the form o f frequen cy distributions as determ ined by the computer.
In addition, both the arithm etic means and the standard deviations for
the various measured quantities w ere determ ined. The frequency distribu­
tions reveal clea rly noticeable m axim a fo r the speed and course values
that in all p robability existed in fact.
Figures 7 and 8 show longitudinal and transversal speed values
respectively. The m easurem ent interval fo r the distribution was 0.02 kt.
T h e effects o f the doppler fluctuation are included in the short-term
fluctuations, but in this case undesired, though small, true acceleration
processes have a m inor effect.
The transversal speed is subject to additional fluctuations as a result
o f course unsteadiness, i.e. if the vessel yaws about its vertical axis,
as this does not necessarily pass through the point at which the transducer
is installed. It can be clearly seen that the fluctuation o f the transversal
speed values is alm ost tw ice as large as that o f the longitudinal values.
In the exam ple given, the short-term fluctuations in longitudinal and
510
512
514
516
518
520
522
524
526
528
530
532
534
536
538
1. 540
542
544
546
548
550
552
554
556
558
560
562
1. 564
566
568
570
572
574
576
578
580
582
584
586
588
L o n g i­
tu d in a l
speed in
k n o ts (1 /1 0 )
++++++++++++
Date o f voyage
1 M arch 1972
R e co rd N o .
7 0 . . . 86
V lm
15.54 kn o ts
°vf
0 .1 0 k t
M easurem ent
p e rio d
8.7 m in
N u m b e r o f values
m easured
320
in te rva l
F i g . 7. — Statistical short-period evalu ation o f the lon gitu din al speed.
(Frequency d is tribu tio n ).
transversal speed are respectively 0.65 % and 1.15 % in relation to the
longitudinal speed. Figure 9 shows short term fluctuations in course
(track angle) values. The interval chosen for the frequ en cy distribution
is 0.05". It is m ainly the actual unsteadiness o f the vessel on its course
that is here reflected.
H i-F ix positions were used to test route accuracy. In order to be
able to differen tiate between longitu dinal and transverse errors, and also
-0 . 062
-0 . 060
-0 . 058
-0.0 5 6
-0.0 5 4
-0.05 2
-0 . 050
-0 .0 4 8
-0 .0 4 6
-0 .0 4 4
-0 . 042
-0 . 040
-0.0 3 8
-0 . 036
-0 . 034
-0 .0 3 2
-0 .0 3 0
-0 .0 2 8
-0 . 026
-0 . 024
- 0.022
- 0.020
-0 .0 1 8
-0 .0 1 6
-0 .0 1 4
- 0.012
- 0.010
-0 . 008
-0 . 006
-0.0 0 4
- 0.002
0 . 000
0 . 002
0. 004
0. 006
0. 008
0 , 010
0,012
0. 014
0. 016
0. 018
0
2 ++
0
0
0
1 +
1 +
1 +
3 +++
0
4
3
2
2
2
9
11
6 ++++++
8 ++++++++
5 +++++
13
9 +++++++++
12 ++++++++++++
9 +++++++++
10
12
10
18
16
16
11
18
18
6 ++++++
7 +++++++
9
8 ++++++++
8
13
6
6
0 . 020
8
3
Date o f voyage
1 M arch 1 972
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
6
1
R e co rd No.
7 0 . . . 86
vqra
-0 .0 6
1
avq
0 .1 8 k t
0
0
0
M easurem ent
p e rio d
8 .7 m in
N u m b e r o f values
measured
320
0 . 022
024
026
028
030
032
034
036
038
040
042
T ransver­
sal speed
in k n o ts (1 /1 0 )
F ig . 8. —
0
4
2 ++
0
N um ber o f
values w ith in
th e speed
in te rva l
S t a t is t ic a l
<v / m
kt
1 5.54 kn o ts)
s h o r t - p e r io d e v a lu a t io n o f th e t r a n s v e r s a l
( F r e q u e n c y d is t r ib u t io n ).
sp eed .
65. 950
66 . 000
66.050
66. 100
66.150
66.200
66.250
66.300
66.350
66.400
66.450
6 6 . 50 0
66.550
66. 6 0 0
66.650
66. 70 0
66.750
66. 8 0 0
66.850
66 . 9oo
66.950
67.000
67. 050
67. 100
67.150
67. 2 0 0
67.250
67. 3 0 0
67.350
67.400
67.450
67.500
67.550
67. 6 0 0
67.650
67.700
67.750
67. 8 0 0
67.850
67. 900
67. 950
D a te o f voyage
1 M arch 1972
R e co rd No.
7 0 . . . 86
tp m
6 6 .9 0 °
a ip
0 .3 0 °
M easurem ent p e rio d
8.7 m in
68 . 000
N u m b e r o f values
68.050
m easured
320
(V ,m
1 5 .5 4 kn o ts)
68 . 100
68.150
Track
angle
values in th e
( °)
tra c k angle
in te rva l
F ig .
9. — Statistical sh ort-period evalu ation o f the track angle.
(Frequency d istribu tion ).
between the system atic and the statistical errors, as w ell as to determ ine
the dependence o f all errors on the distance covered, sections o f route
w ith course and speed as constant as possible w ere utilized fo r those tests.
Judging the reliab ility o f the H i-F ix measurements proved a d ifficu lt
m atter. It was most im portant to avoid including in the evaluations either
any errors due to the H i-F ix itself — as these w ould give the im pression
o f jum ps in position — or any H i-F ix values from unfavourable areas,
i.e. in coastal regions, or where the angles o f intersection o f the hyperbolae
are small. As the H i-F ix values were used for comparison purposes,
a great deal o f im portance was attached to this matter.
There was a further means o f comparison. The vessel proceeded
successively along most o f the sections of route on a constant course and
w ith engine power also constant. As external effects, such as w ind and
current, w ere slight the vessel was able in fact to run at a fa irly constant
ground speed.
In principle, it cannot be expected that in the doppler m easuring
system (w h ich is alm ost w ith ou t inertia) the statistical position errors
w ill compensate fo r the small variations o f speed invariably occurring in
successive route sections to the extent that the resulting speed value w ill
appear more constant than ii rea lly is.
As these measurements w ere carried out w ith constant tim e intervals,
the route sections should also be constant in length and taken at a
constant speed.
A ll the results obtained w ere analyzed to determine the fluctuations
in a series o f successive route sections measured with the doppler and
the H i-F ix systems.
Figures 10, 11 and 12 show some of the results o f this analysis.
AS
i-------------------------------------- 5079,19 m ----------------------------------Date o f voyage
7 Ja n u ary 1972
H i-F ix coverage
K iel lig h th o u s e — O lp e n itz
(stable area)
T o ta l le n g th (H i-F ix )
5 0 7 9 .1 9 m
T o ta l le n g th (D o p p le r) 5 0 9 7 .4 2 m
N u m b e r o f sections
16
Average se ctio n le n g th 3 1 7 .4 4 m
x - - - x - - - x H i-F ix m easurements
Average speed
o ----- o ------ o D o p p le r m easurem ents
1 8 .5 k n o ts
F ig . 10. — Flu ctuations o f H i-F ix and do p p ler measurem ents
along track sections using stable Hi-Fix values.
-394,4 mDate o f voyage
7 January 1972
H i-F ix coverage
B uoy N o . 8a to
Buoy N o. 5a
(in th e w a rn in g region)
T o ta l le n g th (H i-F ix )
3 9 4 .4 m
T o ta l le n g th (d o p p le r)
3 9 8 .2 m
N u m b e r o f sections
8
Average se c tio n le n g th 4 9 .7 7 m
x - - - x - - - x H i-F ix m easurem ents
Average speed
o -----o ------ o D o p p le r m easurem ents
2 .8 9 knots
F ig . 11. — Fluctuations o f Hi-l*'ix and doppler measurements
along track sections using unstable H i-Fix values.
In the examples given, the overall lengths of route w ere determ ined
from their series o f measured sections. Assum ing a constant speed fo r
each overall route length, each o f these was divided by its number o f
sections to obtain an average length for its sections. The values determ ined
from the H i-F ix data were plotted on the abscissae in metres. The
ordinates represent the deviations o f the H i-F ix and the doppler lengths
fo r each successive route section and are also given in metres.
Figure 10 shows the results o f measurements w ith stable H i-F ix
values. The measurements were made over a route from K iel Lighthouse
to the No. 3 O lpenitz buoy. The fluctuations in the values obtained with
H i-Fix and w ith doppler were negligible, being only about ± 1 m ( ± 0.3 % ).
116
A
IN T E R N A T IO N A L
H Y D RO G RAPH IC
R E V IE W
S
---------------------------------- 2039,4, m
Date o f voyage
-----------------------------------------— j
7 January 1972
H i-F ix coverage
B uoy N° 5a (in the w a rnin g
region)
T o ta l le n g th (H i-F ix )
2 0 3 9 .4 m
T otal le n g th (D o p p le r) 2 0 7 2 .6 m
N u m be r o f sections
8
Average section le n g th 2 5 4 .9 3 m
x . . . x . . . x H i-F ix measurements
Average speed
o ------ o -------o D o p p le r m easurem ents
F ig .
15.6 k n o ts
12. — Flu ctuations o f H i-Fix and d o p p le r measurements
along track sections using unstable H i-Fix values.
W h a t is interesting about this exam ple is
strong and obvious correlation between
measurements, indicating the fluctuations
over the overall length o f the route, and in
tions in measurements which rem ain when
considerably smaller.
that it showed that there is a
the doppler and the H i-F ix
o f true speed which occurred
consequence the actual fluctua­
this has been allowed for w ere
Figures 11 and 12 show exam ples o f unstable H i-Fix values. The
measurements w ere taken over the route from Buoy No. 8a to E ckernfôrde
— that is, w ell w ithin the E ck ern ford er Bucht. The fluctuations in H i-F ix
values w ere clearly larger than those of the doppler values w hich w ere
once more in the range o f ± 1 m. No correlation exists between the
deviation values in the tw o systems. H i-Fix measurements o f this kind
cannot be used as com parative data, and were not used for the analysis.
A com puter program was used to calculate the length o f each section
between the starting and end points o f the route and their northreferenced course, in order to determ ine the route errors fro m the doppler
and H i-F ix positions.
Then the differences between the doppler and the H i-F ix route lengths
(h erein after called “ longitudinal route d iffe r e n c e s ” ) w ere determ ined,
together w ith the course differences. The “ transversal route differences ”
could then be calculated from the course differences by being referenced
to the H i-F ix route length. Systematic and statistical components w ere
determ ined from the longitudinal and transversal differences fo r route
sections w ith approxim ately constant length. F in ally, the mean arithm etic
errors and the mean square errors were calculated as a function o f the
length o f the section o f route. About 460 sections of route w ere analysed
from three typical test runs which it should be noted included manœuvres
in volvin g great changes in course as well as o f speed (up to ± 22 knots).
This fact is o f importance, since the acceleration error o f the gyro compass
has larger effects under such conditions.
The longest route sections with constant course and speed w ere of
5 nautical miles, and a number o f them w ere m onitored with the H i-F ix
system and then analysed. It was established that most o f the course
errors that had been assumed to be statistical, resulting from gyro compass
acceleration errors over successive sections o f the route, are in fact
correlated. T h is fact must be taken into consideration when analyzing
the results.
Fgures 13 to 16 show the results, and take account
for both the doppler system and the H i-F ix system.
o f the errors
Figu re 13 gives inform ation about the characteristics o f the system atic
longitudinal error which to begin w ith had been deliberately left un­
corrected. W ith increasing distance, the erro r increases in a nearly linear
way. The relative systematic error for the route is approxim ately —
|—0.3 %.
On the basis o f these measurements a correction factor o f 0.97 was fed
into the com puter in order to elim inate this system atic error o f the system.
Figu re 14 demonstrates the standard deviation o f the overall lon gi­
tudinal route error, that is to say, the statistical error.
T h e dashed line indicates ap proxim ately the curve o f the theoretical
function. W h a t is im portant is that the increase in the accidental error
for the longitudinal route is not proportional to the distance covered
but only to the square root o f this distance. A ccordin g to this figure the
relative standard deviation is approxim ately o f ± 1 to ± 0.8 % over short
distances, ± 0.4 to ± 0.3 % over medium distances and ± 0.18 % over the
longest section o f route. Th e relative error w ould decrease even further
over longer distances.
In those sections o f route o f up to 5 nautical miles in length the curve
flattens out and this demonstrates that the errors occurring in the
successive sections are rem arkably independent from a statistical point
o f view.
The curve that m ay be theoretically expected w ill not pass through
Ihe origin o f coordinates, but w ill cut the axis o f the ordinate at a point
Fir.. 1.1 — Systematic
measurement error j : the Alp1’ » Popper
test system for straight routes.
.Navigator
10 V
\o —J
+1
e
Fir.. 14. —
RMS
m easurement error of the Alpha
on straight routes.
Doppler
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cn
Fig. 16. — RMS residual error for the com puter-corrected gvro course during
manœuvres
(circular, triangular
and
oscillating
runs
in North-South
West directions with
speeds varying up to ± 22 k n o ts).
O
o
-o
o
ap proxim ately ± 2 m. This value corresponds approxim ately to the
H i-F ix system error, which is independent o f the route sections and is
superimposed on all the measured values.
Figu re 15 shows the system atic transversal error versus the distance
covered.
The relative error amounts to approxim ately 0.65 %, and
corresponds to a system atic course error o f 0.37°. The course was
corrected in the com puter program by — 0.37° and thus the error was
on the average eliminated.
F igu re 16 shows the standard deviation o f the statistical transversal
route error measured during the ship’s most com plicated manœuvres.
F o r this range o f short sections, unlike the longitudinal error, the
transversal error shows an increase proportional to the distance covered.
In actual fact the course errors in the successive sections — errors which
were assumed to be accidental — are still very closely correlated since
the period o f the oscillations of the acceleration error exceeds 1 hour.
The relative error amounts to ap proxim ately ± 0.85 %, thus corresponding
to 0.5°. This being so, the overall accuracy o f the system depends
essentially on the accuracy o f the course. It can be expected that this
acceleration error w ill decrease considerably for slow er speeds than those
em ployed during the test, and fo r less extrem e changes o f course. Even
here a sm oothing effect w ill m anifest itself after the run has been in
progress for more than several hours, that is to say the absolute
transversal error w ill increase b y approxim ately the square root o f the
distance covered, and the relative transversal error w ill decrease by the
inverse o f this square root. The graph in figure 16 is for the area
im m ediately surrounding the origin o f coordinates shown in figure 14.
Here, too, the expected theoretic curve w ill not pass through the origin
o f coordinates but w ill cut the axis o f the ordinates at a point about ± 2 m
from the origin. This w ill constitute the approxim ate standard deviation
— w h ich w ill be superim posed on all measurements — o f the statistical
error which is independent o f distance. This error is not caused by the
doppler system.
T h e overall accuracy o f the system can finally only be determ ined
by means o f these statistical errors o f route and course. For a 50 % mean
circu lar error probability (C E P ) a value o f 0.41 % was determ ined by
analyzing 22 test runs, all p erform ed during com plicated manœuvres,
each test varyin g between 20 and 90 nautical miles, the total length being
1200 nautical miles. T h is relative value refers to the distance covered,
and the percentage o f error is certain to be less for longer routes and lower
speeds, as w ell as for less frequ en t alterations o f course and speed.
T h e determ ination o f the correction values for longitudinal speed
and fo r the course, i.e. the actual corrections to the system, m ay in practice
be perform ed as follow s :
T h e ship covers a distance I between tw o geographically known
points A and B. This distance should be as long as possible (see fig. 17).
T h e uncorrected doppler m easurem ent leads to a point B'. Deviation
A I fo r the longitudinal direction o f the route and deviation A q fo r the
transversal direction can then be determ ined.
Thus (1 ± A l / l ) is the correction factor for the distance and speed
measurements, and Aq> = arctan A q / l the correction factor for the compass
error.
Since in the case of a single measurement the statistical errors are
included in the measurement results the correction values obtained must
be regarded as first approximations. T h e y may be im proved by further
test runs with any known starting and finishing points, thus without the
necessity of follow in g a particular route.
In conclusion, and to summarize, a few remarks should be added.
This article has attempted to show the com plexity of the problem of
environmental effects on the accuracy o f position fixin g with the sonar
doppler system.
In this connection, it is im portant to have the possibility both
o f knowing the effects of a large number of possible errors and of
elim inating them by physical or mathematical means. O nly a part of
the statistical errors o f course and speed then remain as position errors.
Analysis o f these rem aining position errors reveals the problem o f the
accuracy of the reference system. It is only by spending much time
(for many long distances have to be covered) and by the use o f the most
accurate known surveying methods that it is possible to give firm evidence
o f the errors attributable to the sonar doppler system.
W hen discussing the results obtained from measurements account
must be taken of the fact that the test runs w ere planned so as to represent
nearly the worst possible conditions for the gyro compass. The resulting
errors are rem arkably small. Furtherm ore, the relative errors show a
tendency to decrease with the length o f the route.
As relative error values are bound to continue to decrease with lower
speeds (w ith the ship Biirkner this was not possible), and w ith less frequent
changes of course and speed, it can be asserted that this navigation system
consisting o f two doppler components and a com mercial compass opens
up new possibilities of activity fo r hydrographic surveying as w ell as for
the accurate navigation of commercial vessels in coastal waters.