Altimetry method

An altimetry method uses one or more sources of opportunity, for example transmitters on satellites of the "GPS" satellite navigation system. One or more receivers on board an aircraft or a satellite in low Earth orbit are used. Multiple correlation is applied between the direct signal received from the transmitter and the reflected signal. The coordinates of the point of specular reflection are derived from the measured delay achieved by said multiple correlation and comparison with a theoretical model of the terrestrial sphere. The device for implementing the method includes a signal processor, a variable delay circuit, a discrete delay line, a bank of correlators and detectors. A specific application of the method and device is ocean altimetry.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention concerns altimetry from the air or from space, 
especially ocean altimetry. 
The invention also concerns a device for implementing the method. 
2. Description of the Prior Art 
As recognized during the Consultative Meeting on Imaging Altimeter 
Requirements and Techniques held in June 1990 at the Mullard Space Science 
Laboratory, the ability to carry out high precision ocean altimetry over a 
swath with a high spatial resolution would revolutionize many fields of 
earth science. 
There are many methods for carrying out ocean altimetry. The three main 
parameters associated with these methods and defining their respective 
performance are: vertical precision, spatial resolution and swath. 
Most conventional methods are based on the use of radar emitting signals of 
suitable wavelength, for example an airborne radar. There are various 
types of radar: single-pulse system, interferometric system, etc. 
Altimetry of this type has been mostly limited to nadir-looking type 
instruments, restricting the range of possibilities. 
Some prior art methods used existing radio frequency signals and combine in 
an airborne or space receiver signals from the transmitter (direct 
signals) and signals reflected from the ocean (or, more generally, from 
the terrestrial surface). In the following description the expression 
"terrestrial surface" refers to the terrestrial crust, or to the surface 
of oceans, seas or lakes, or to the surface of frozen water. 
The following two documents described such methods: 
The article by Philipp HARTL and Hans Martin BRAUN: "A Bistatic Parasitical 
Radar (BI)" published in "INTERNATIONAL ARCHIVES OF PHOTOGRAMMETRY AND 
REMOTE SENSING", vol 27, 1988, pages 45-53; and 
the technical report "AIRCRAFT ALTITUDE DETERMINATION USING MULTIPATH 
INFORMATION IN AN ANGLE-MEASURING NAVIGATION SATELLITE SYSTEM" by David 
KURJAN, Moore School Report no. 72-12 of 30 Sep. 1971. 
Usable existing sources of radio frequency signals, known as "sources of 
opportunity", include communication and television satellites such as the 
"GPS" ("Global Positioning Satellite") satellite navigation system used in 
the West and its equivalent GLONASS in the former Soviet Union. 
Whilst retaining the concept briefly mentioned above (use of sources of 
opportunity and combination of direct and reflected waves), this invention 
is directed to providing a very high precision altimetry method with a 
specific application to estimating the height of seas of oceans and 
variations therein. This invention is also directed to maintaining this 
precision over large area swaths, the measurements no longer being 
restricted to the nadir. 
To this end, one major feature of the method of the invention is the use of 
multiple correlation of the direct and reflected signals. 
For simplicity, but without any intention of limiting the invention, the 
remainder of this description concerns signals from "GPS" system 
satellites and a receiver on a satellite in low earth orbit ("LEO"). The 
receiver can be an airborne receiver without departing from the scope of 
the invention. 
Also for reasons of simplicity, unless otherwise indicated the remainder of 
this description concerns only specular reflection. Signals obtained by 
diffuse reflection can nevertheless be used, especially in sea ice mapping 
applications. 
To be more precise, altimetry is performed by measuring accurately the 
coordinates of the point of specular reflection by delay measurements and 
using a geodetic Earth model. 
One model that can be used is the WGS-84 ("World Geodetic System 1984) 
model. A definition of the WGS-84 model can be found in ANON, "Department 
of Defence World Geodetic 1984--its definition and relationships with 
local geodetic systems", Defence Mapping Agency Technical Report, No. DMA 
TR 8350.2, second edition, 1988. 
A general description of geodetic systems is given in the article by P.A. 
CROSS: "Position: Just What Does it Mean!", published in "Proceedings of 
NAV'89 Conference, Royal Institute of Navigation, London. 
Finally, if the transmitter is one of the "GPS" satellites or a source of 
signals of this type, its instantaneous coordinates and the 
characteristics of the signals transmitted are accurately known. 
Until now it has been implicitly assumed that a single source of signals is 
used, of example the transmitter of one of the "GPS" system satellites. A 
plurality of spatially separate sources can advantageously be used. In 
this case altimetry can be carried out over several subtracks which will 
be spread over a distance depending on the receiver altitude. 
If several receivers are assumed, as in the case of a constellation of 
small receiving satellites, then the subtrack distance can be determined 
by adjusting the orbits of the receivers. 
The invention also authorizes real time processing of the data using 
on-board signal processing devices, necessary for implementation of the 
method, unlike prior art systems (using "SAR" type radars defined below, 
for example) which use installations on the ground to process the acquired 
data. This leads to significant delays in availability. 
SUMMARY OF THE INVENTION 
The invention includes an altimetry method using sources of opportunity, 
i.e. signals transmitted by at least one transmitter on board a platform 
in terrestrial orbit with particular characteristics; the method 
comprising at least a stage of receiving and combining, in a receiver on 
board a platform in terrestrial orbit with particular specifications or on 
board an aircraft, signals received direct from said transmitter and 
signals reflected from the surface of the terrestrial sphere, and a stage 
of measuring the coordinates of the point of specular reflection of the 
signals transmitted from the surface of the terrestrial sphere by 
measuring the propagation delay of the reflected signals, a stage of 
comparing with a theoretical model of the geometrical properties of the 
terrestrial sphere, in order to determine variations in altitude of said 
terrestrial surface relative to said model, and in which method said 
combination stage, to obtain the echo waveform, entails multiple 
correlation of the received reflected signal with a particular number of 
replicas delayed by a constant amount of the received reflected signal, 
said waveform having an upstream noise floor and a downstream pulse of 
entirely reflected power amplitude bracketing a median range representing 
the delay associated with said point of specular reflection. 
The invention also includes an altimetry device implementing this method. 
The invention will be more clearly understood and other features and 
advantages of the invention will emerge from a reading of the following 
description given with reference to the appended drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
As already mentioned, it is first assumed that there is one source of 
opportunity, i.e. a single transmitter E and a single receiver R. FIG. 1 
shows this situation. 
One problem to be overcome is that of determining precisely the coordinates 
of the point P.sub.S of specular reflection on the terrestrial surface T 
of a wave emitted by the source E and picked up by the receiver R. 
Initially the Earth is assumed to be a perfect sphere of radius r and 
center o. 
Let xy be reference plane such that the plane xOy contains the transmitter 
E, the receiver R and a reflection point P. The cartesian coordinates of P 
can be expressed by the function P (x, y), those of R by the function R 
(x, y) and those of E by the function E (x, y). The polar coordinates of P 
are the angle .phi. relative to the Ox axis and the terrestrial radius r. 
Changing coordinates and choosing a new reference plane x'Oy' such that the 
axis Ox' is coincident with the line bisecting the angle 
RPE=2.times..theta. and the origin is the point P itself (translation by a 
distance r), the relationship between xy (the coordinates of a point in 
the original reference plane) and x'y' (the coordinates of a point in the 
new reference plane) is then as follows: 
##EQU1## 
The coordinates of the transmitter E and the receiver R are transformed 
into the new system of reference: 
##EQU2## 
Because the reflection is specular reflection, the following equation is 
satisfied: 
##EQU3## 
Carrying out the standard substitutions and the following change of 
variable: 
EQU t=tan .phi./2 (5) 
the equation of the specular reflection point P.sub.S can be written as 
(spherical mirror equation): 
EQU c.sub.4 t.sup.4 +c.sub.3 t.sup.3 +c.sub.2 t.sup.2 +c.sub. t+c.sub.0 =0 (6) 
where: 
EQU c.sub.0 =(x.sub.e y.sub.r +y.sub.e x.sub.r)-r(y.sub.e +y.sub.r)(7) 
EQU c.sub.1 =-4(x.sub.e x.sub.r -y.sub.e y.sub.r)+2r(x.sub.e +x.sub.r)(8) 
EQU c.sub.2 =-6(x.sub.e y.sub.r +y.sub.e x.sub.r) (9) 
EQU c.sub.3 =4(x.sub.e x.sub.r -y.sub.e y.sub.r)+2r(x.sub.e +x.sub.r)(10) 
EQU c.sub.4 =(x.sub.e y.sub.r +y.sub.e r.sub.r)+r(y.sub.e +y.sub.r)(11) 
It is also necessary to determine the distance from the receiver R to the 
point P.sub.S of specular reflection as a function of the elevation of the 
transmitter E above the local horizon. It is assumed arbitrarily that the 
receiver R is located along the Oy axis of an orthonomic trihedron yOx at 
an altitude of h over terrestrial surface T (see FIG. 2). Its coordinates 
are: 
EQU x.sub.r =0 and y.sub.r =r+h (12) 
The coordinates of the transmitter E can be expressed in terms of the 
elevation angle .beta. above the local horizon at the location of the 
receiver R using the following relations, as directly derived from FIG. 2: 
##EQU4## 
where r.sub.e is the orbital radius of the transmitter E and .gamma. is 
the angle REO. Conventional manipulation yields the required coordinates: 
##EQU5## 
Let R.sub.2 =RP0 be the distance between the receiver R and the point 
P.sub.S of specular reflection and S the arc length from the point of 
projection P.sub.R of the receiver R onto the terrestrial surface T: 
EQU =(.pi./2-.phi.)r (16) 
If the transmitter E is one of the "GPS" system satellites, the orbital 
radius is r.sub.e =26.5 10.sup.6 m. The value r=637 10.sup.6 m is taken 
for the Earth radius. Using these values and the above formula, the arc 
length S and the distance R.sub.2 are readily calculated from the 
elevation .beta. of the transmitter E above the local horizon at the 
position of the receiver R and from h. 
For example, consider three values of the altitude h: 
a/ h=700 km (low earth orbit satellite); 
b/ h=10 km (aircraft); and 
c/ h==1 km (low-flying aircraft). 
The arc length S and the distances R.sub.2 are calculated for variations in 
the angle .beta. between 0.degree. and 90.degree.. The results of these 
calculations are set out in tables 1 through 3 in APPENDIX 1 to this 
patent application. 
It is necessary to define an additional parameter: the iso-range lines. The 
iso-range lines are defined as those points for which the relative delay 
between the direct signal and the reflected signal is the same. This 
condition can be written, with reference to FIG. 3: 
EQU EP+PR-ER=K (17) 
where E is the transmitter point, R the receiver point and P the reflection 
point for which the path difference is K (in meters if the other distances 
are expressed in this unit). It is assumed that the propagation time 
depends essentially on the distance and not on the medium. 
It can also be assumed that the positions of the transmitter E and the 
receiver R are known from the information provided by the transmitted 
signals themselves and that therefore the distance ER is a known constant 
in the above equation. For this reason the following equivalent expression 
can be written: 
EQU EP+PR=k (18) 
with k being a constant equal to: 
EQU k=K+ER (19) 
The above equation is of an ellipsoid of revolution with foci at the 
transmitter point E and at the receiver point R as depicted in FIG. 3. The 
intersection of this ellipsoid and the terrestrial surface T defines a 
particular iso-range line L.sub.isoP. A family of iso-range lines 
L.sub.iso is generated when k is allowed to take different values. 
The instrument of the invention measures the quantity k. A known range 
processor uses several values of this delay separated by one segment of 
the transmitted signal ("code chip") to perform range discrimination. The 
signals transmitted by "GPS" type satellites are not made up of binary 
words with a regular structure, but rather of pseudo-random bit sequences. 
In the following description, the expression "code chip" is used for the 
specific signal just defined. To find the width of the strip on the ground 
corresponding to this "code chip" it is necessary to know the gradient of 
k as a function of the point P.sub.S of specular reflection. The equation 
relating the spatial sensitivity k.sub.S of the instrument and the point 
P.sub.S is as follows: 
##EQU6## 
where s is the coordinate (arc) measured along a coordinate line 
perpendicular to the iso-range line. 
In general, the intersection of the ellipsoids defined by equation (18) 
above and the terrestrial surface (modelled also as an ellipsoid of 
revolution) gives a curve which is not flat. 
Consider first the general case where the transmitter E is in any arbitrary 
direction. The particular case where the transmitter E is in the zenith 
direction of the receiver R is considered later. For this particular 
configuration the iso-range lines and the sensitivity are calculated. The 
Earth with always be approximated by a sphere. 
The general case is shown in FIG. 4. For the coordinates an Earth centered 
coordinate system is chosen such that the receiver R is on the Ox axis and 
the transmitter E is in the xOy plane. The coordinates of the points E, R 
and P satisfy the following equations: 
EQU E=(x.sub.ey.sub.e O); R=(x.sub.r,O,O); P=(x,y,z) (21) 
The ellipsoid defined by equation (18) can be written as: 
##EQU7## 
and the equation for the terrestrial sphere as: 
EQU x.sup.2 +y.sup.2 +z.sup.2 =r.sup.2 (23) 
The iso-range lines l.sub.iso are defined by the intersection of the 
ellipsoids and the terrestrial sphere. When this intersection is 
calculation the following equation is found for the projection of the 
iso-range lines I.sub.iso onto the xOy plane: 
EQU ax.sup.2 +by.sup.2 +cxy+dx+ey+f=0 (24) 
The coefficients "a" through "f" are easily calculated from the equations 
(21) through (23). 
Consider now the case where the transmitter E is in the zenith direction of 
the receiver R, that is the case for which y.sub.e =0. 
Equation (24) can be rewritten as follows: 
EQU Ax.sup.2 +Bx+C=0 (25) 
with: 
EQU A=4(x.sub.e -x.sub.r).sup.2 (26) 
EQU B=-4(x.sub.e +x.sub.r)[(x.sub.e -x.sub.r.sup.2 -k.sup.2 ] (27) 
EQU C=(x.sub.e x.sub.r).sup.2 (x.sub.e -x.sub.r).sup.2 -2k.sup.2 (2r.sup.2 
+x.sub.e.sup.2 +x.sub.r.sup.2)+k.sup.4 (28) 
The equations of the iso-range lines l.sub.iso are finally obtained: 
EQU y.sup.2 +z.sup.2 =r.sup.2 -x.sup.2 (k) (29) 
with 
##EQU8## 
The iso-range lines are thus circles parallel to the yOz plane and centered 
on the Oz axis, as shown in FIG. 5a. The range of values of k extends from 
a minimum value at the point of projection of the receiver R onto the 
terrestrial surface T to a maximum value at the point of tangency to the 
terrestrial sphere. These two limits are given by the following equation: 
##EQU9## 
To compute the spatial sensitivity of the instrument of the invention, 
equation (28) can be rewritten as follows: 
EQU k.sup.4 -2Dk.sup.2 +E=0 (31) 
with: 
EQU D=(x.sub.e -x).sup.2 +(x.sub.r -x).sup.2 +2(r+x)(r-x) (31a) 
and 
EQU E=[(x.sub.e -x).sup.2 +(x.sub.r -x).sup.2 ].sup.2 (32) 
The equation finally obtained for the sensitivity is: 
##EQU10## 
The x coordinate is related to the length of the arc s measured from the 
point of projection of the receiver R onto the terrestrial surface T along 
any great circle. The value of x then satisfies the following equation: 
EQU x=r cos (s/r) (34) 
Equation (33) above can be rewritten in the following manner as a function 
of s: 
##EQU11## 
The arc length coordinate lines are perpendicular to the iso-range lines. 
The spatial sensitivity of the instrument is therefore given by the 
following equation, after differentiating equation (35): 
##EQU12## 
with 
##EQU13## 
The arc length s extends from the point of projection of the receiver E 
onto the terrestrial surface T to the point of tangency. The limits of 
this arc length can easily be calculated from the following relation: 
EQU 0.ltoreq.s.ltoreq..gamma.arc cos .gamma./x.sub.c (39) 
The spatial sensitivity of the instrument has been calculated for various 
arc lengths in the case of a low earth orbit satellite. The results are 
shown in the table below, along with the spatial resolution obtained using 
a "GPS" system satellite and the C/A and P codes, respectively. 
______________________________________ 
s (km) 
ks = ds/dk (m/m) .rho.g C/A (m) 
.rho.g P (m) 
______________________________________ 
0 .infin. .infin. .infin. 
10 62 18600 1960 
100 6 1800 180 
200 3.2 960 96 
500 1.4 420 42 
800 1.1 330 33 
1100 0.9 270 27 
3030 0.6 180 18 
______________________________________ 
This table shows clearly that the sensitivity is better for longer 
distances from the receiver R to the target. The best sensitivity is 
obtained at the point of tangency, and is about 0.6. Near the sub-receiver 
point the sensitivity is very poor (k.sub.s tends to infinity). For ranges 
corresponding to arc lengths s greater than approximately 200 km the 
sensitivity is better than 3. 
Detailed consideration has just been given to the specific case in which 
the transmitter E is at the zenith of the receiver R. The point P.sub.S of 
specular reflection coincides with the projection of the transmitter E 
onto the terrestrial surface. The iso-range lines l.sub.iso are circular 
and concentric with the point P.sub.S of specular reflection. 
When the transmitter E is at an elevation other than zenith, the specular 
reflection point is on the terrestrial surface between the receiver R and 
the transmitter. In this case the iso-range l-iso will also be concentric 
around the point P.sub.S of specular reflection but will be ellipses. The 
separation between them decreases in the direction away from the point 
P.sub.S. This configuration is shown in FIG. 5b. The point P.sub.S of 
specular reflection is disposed away from the point P.sub.R of projection 
of the receiver R onto the terrestrial surface and towards the transmitter 
E (on the right of the figure in the example shown). 
Another phenomenon has to be taken into account: the Doppler effect. The 
transmitter R moves in space at a particular speed, which gives rise to 
the Doppler effect. The iso-Doppler lines can be determined in a similar 
manner to the iso-range lines. 
Refer now to FIGS. 6 and 7. 
From the FIG. 6 vector diagram, the relative radial velocity for the signal 
received through the direct path is: 
EQU v.sub.1 (v.sub.r -v.sub.c)u.sub.re (40) 
where: 
v.sub.1 is the modulus of the radial velocity between the transmitter E and 
the receiver R along the direct path; 
v.sub.c is the velocity vector of the transmitter E; 
v.sub.r is the velocity vector of the receiver R; 
u.sub.re is the unit vector pointing from the receiver R towards the 
transmitter; and 
a dot (.) indicates the scalar product of vectors. 
The sign has been chosen such that a positive radial velocity means that 
the transmitter E and the receiver R are moving towards each other. 
Likewise, the receiver radial velocity for the signal received through the 
reflected (indirect) path is: 
EQU v.sub.2 =(v.sub.r -v.sub.P).u.sub.rP +(v.sub.P -v.sub.e).u.sub.P6(41) 
where: 
V.sub.2 is the modulus of the radial velocity between the transmitter E and 
the receiver R along the reflected (indirect) path; 
v.sub.P is the velocity vector of the target; 
v.sub.rP is the unit vector pointing from the receiver R towards the 
target; and 
u.sub.pe is the unit vector pointing from the target towards the 
transmitter E. 
The difference in radial velocity between the direct and reflected 
(indirect) paths is given by the equation: 
EQU .lambda.f.sub.D =v.sub.2 -v.sub.1 (42) 
whence 
EQU .lambda.fD=(v.sub.r- v.sub.P).u.sub.rP +(v.sub.p -v.sub.e).u.sub.Pe 
-(v.sub.r -v.sub.e).u.sub.re (43) 
This expression can be simplified by assuming that the distance between the 
receiver R and the target is negligible in comparison with that between 
the transmitter E and the receiver R or the target, whence: 
EQU u.sub.re .apprxeq.u.sub.Pe (44) 
In this case, the above equation (43) can be written: 
EQU .lambda.f.sub.D =(v.sub.r -v.sub.P).u.sub.rP (v.sub.P -v.sub.r).u.sub.re(45 
) 
The first term in the righthand part of this equation is similar to that 
encountered in slant SAR (Synthetic Aperture Radar), except for a factor 
of 2 due to the monostatic mode of operation. The second term depends only 
on the velocity v.sub.P of the target. As the target velocity is due to 
the rotation of the Earth which is the same for all points in the 
footprint, to a first approximation, it can be considered that the second 
term is a known constant over the antenna footprint of the receiver R. 
If the rotation of the Earth is neglected, i.e. if v.sub.P =0, then 
equation (45) becomes: 
EQU .lambda.f.sub.D =v.sub.r.u.sub.rP -v.sub.r u.sub.re (46) 
As the second term on the righthand side is assumed to be a known constant, 
the iso-Doppler lines are contained in the following family of cones: 
EQU v.sub.r.u.sub.rP =k.sub.D (47) 
where: 
EQU k.sub.D =.lambda.f.sub.D +v.sub.r.u.sub.re (48) 
Intersection of these cones and the terrestrial surfaces gives the 
iso-Doppler lines l.sub.Dop. If the terrestrial surface is locally 
approximately by a plane, the well known family of hyperbolic lines used 
in conventional slant SAR is obtained, as shown in FIG. 7. Note, however, 
that although the pattern is the same, the value of the Doppler frequency 
for each curve will have a constant difference with respect to the usual 
SAR situation. 
The conclusion is that, to a first approximation, the pattern of 
iso-Doppler lines l.sub.Dop is the same as the similar curves obtained for 
slant SAR. The small difference is explained by the radial dispersion of 
velocities over the footprint relative to the transmitter, which is 
estimated to be in the order of 50 m/s for points 400 km apart. 
A further difference with respect to monostatic slant SAR is that the 
families of iso-range and iso-Doppler curves can be oriented in different 
directions, depending mainly on the direction of the transmitter E and the 
velocity of the receiver R. This is illustrated in FIG. 7 where the 
transmitter E is assumed to be located along the Ox axis while the 
receiver R is moving in a direction 45.degree. away from this axis. The 
iso-range lines appear oriented towards the transmitter E while the 
iso-Doppler lines are oriented along the direction of motion of the 
receiver R. 
A final parameter has to be taken into account: it is the "signal to noise" 
ratio SNR.sub.0. The following multistatic radar equation for a 
distributed target, familiar to the person skilled in the art and 
therefore needing no detailed description, is used for this. With 
reference to the power of the source, a transmitter E on a "GPS" system 
satellite is used. The general equation for single-pulse multistatic radar 
is: 
##EQU14## 
where: 
SNR.sub.0 is the signal to noise ratio, for a single pulse; 
P.sub.e is the transmitted power (by the "GPS" satellite, for example): 
G.sub.e is the antenna gain of the transmitter E; 
R.sub.1 is the means distance from the transmitter E (the "GPS" satellite) 
to the receiver antenna footprint of the receiver R on the terrestrial 
surface; 
.sigma..sub.b is the mean normalized bistatic radar cross-section across 
the receiver antenna footprint, evaluated at the directions of the 
transmitter E and the receiver R; 
A is the area of the receiver antenna footprint on the terrestrial sphere; 
R.sub.2 is the mean distance from the receiver R to the footprint of the 
transmitter E on the terrestrial surface; 
.lambda. is the wavelength used; 
G.sub.r is the antenna gain of the receiver R; 
K.sub.B is Boltzmann's constant; 
T.sub.s is the system temperature, including thermal noise from the scene 
and receiver noise; and 
B is the bandwidth of the signal. 
These various parameters are shown in FIG. 8. 
The system temperature T.sub.S includes the temperature T.sub.a of the 
antenna of the receiver R due to thermal noise received from the observed 
scene and also the equivalent temperature T.sub.eqr due to noise generated 
by the receiver R itself. The equivalent temperature T.sub.eqr of the 
receiver R can be expressed in terms of the noise figure F of the 
receiver, by the following equation: 
EQU T.sub.eqr =(F=1)T.sub.0 (50) 
where T.sub.0 =290.degree. K. 
Various conventional approximations can be applied, in particular allowing 
for temperatures usually encountered in this kind of application and 
because this is a "pulse limited" system. The dimensions of the footprint 
are very small compared to the distances between the target and the 
transmitter E or the receiver R. Under these conditions, the normalized 
bistatic radar cross-section averaged over the antenna footprint of the 
receiver R can be written as: 
EQU .sigma..sub.b =a/A.sigma..sub.0 (0) (50a) 
where .sigma..sub.0 (0) is equal to the normalized monostatic radar 
cross-section .sigma..sub.0 at .degree. incidence and the "signal to 
noise ratio" satisfies the simplified equation: 
##EQU15## 
In accordance with one aspect of the invention the reflected signal is 
correlated with the direct signal in order to perform the altimetry 
measurement. Each correlation is carried out over the duration of N "code 
chips". As the correlation process is performed coherently an improvement 
is achieved over the single-pulse "signal to noise ratio". The expression 
for the "signal to noise ratio" therefore becomes: 
EQU SNR.sub.N =SNR.sub.0 .times.N (52) 
where SNR.sub.0 is the "signal to noise ratio" for one pulse and N is the 
number of pulses. 
FIG. 9 shows the motion at the velocity v of the receiver R relative to the 
target, the latter being the pulse limited footprint of the area a. 
By applying various standard approximations, and in particular by 
neglecting in the ocean altimetry application the speed at which the 
surface of the ocean moves relative to the transmitter, the number of 
pulses which can be added coherently is given by the following equation: 
##EQU16## 
in which .rho..sub.a is the length of the pulse limited footprint along the 
half-mirror axis and R.sub.2 is the distance from the receiver R to the 
target. 
The coherence time is given by the following equation: 
##EQU17## 
By way of example, table 4 in APPENDIX 2 at the end of this patent 
application sets out the results obtained in the case of a "GPS" system 
satellite for codes transmitted in "C/A" and "P" modes by the transmitters 
of these satellites. The receiver R is a satellite in low earth orbit 
("LEO") at an altitude of 700 km. The "chip rate" is 1 MHz for "C/A" 
("Coarse Acquisition") pseudo-random codes or 10 MHz for "P" ("precision") 
codes. The values given in the table are given by way of example only and 
are in no way limiting on the scope of the invention. 
Having now explained the data to be determined in the context of the 
invention, a detailed description will be given of the features specific 
to the method of the invention. 
As already mentioned, the iso-range lines l.sub.iso are defined by the 
intersection of an ellipsoid whose foci are the receiver R and the 
transmitter with the ellipsoid of the terrestrial sphere T. In the context 
of the present invention, a model is chosen to represent this ellipsoid. 
The WGS-84 model referred to in the preamble to the description is 
preferably chosen. To simplify the following description only this model 
is referred to. However, other models can be used. FIG. 10a illustrates 
the WGS-84 ellipsoid and several iso-range lines: t=0, .tau., .tau.2, 
.tau.3. Considering a specific "code chip", the first echo of the same 
"code Ship" received after reflection at the ocean surface (more generally 
at the surface of the terrestrial sphere) corresponds to the delay 
associated with the ellipsoid tangential to the terrestrial surface. This 
is due to a geometrical property of ellopsoids whereby for any paid of 
straight lines starting from any point P on the ellipsoid and each passing 
through one focus (E and R), the angles .theta.1 and .theta.2 between 
these lines and the normal H at the point P are equal: 
.theta.1=.theta.2=.theta.. This property could be summarized as follows: 
at the point P.sub.S of specular reflection the normals to the iso-range 
ellipsoid and to the ellipsoid representing the terrestrial sphere are 
coincident. 
In the preferred application of the invention, to ocean altimetry, the 
first echo of this same "code chip" will be due to reflection from the 
wave crests. As the delay is increased more reflecting surfaces will 
contribute to the signal received, up to a point at which the last edge of 
the "code chip" reaches a wave trough. Beyond this point the area 
illuminated by the "code chip" remains about the same and the amplitude 
will decrease only because of the shape of the antenna pattern, if it is 
narrow enough, or because of the cross-section of the radar beam with the 
angle of incidence increasing. This takes into account the theory of 
bistatic radar. 
The average shape of the resulting waveform is given by the convolution of 
the response at the target point of the system, the ocean surface height 
distribution and the calm sea impulse response. The antenna footprint is 
usually much larger then the "pulse limited" footprint. 
The illuminated areas for various "code chips" and the average shape of the 
received waveform are shown in FIGS. 10b and 10c. In FIG. 10b the curve 
M.sub.e represents a rough sea and the curve N.sub.moy a calm sea (mean 
sea level). The curve l.sub.isox is one of the iso-range lines from FIG. 
10a. It is easily understood that when the sea is calm, before time t=0, 
the reflected signal has a theoretically null amplitude, and in reality a 
very low amplitude, as shown in FIG. 10c; curve Co.sub.1. This is 
obviously not so for a rough sea (FIG. 10c: curve Co.sub.2). Depending on 
the SWH ("Significant Wave Height" value, signals are reflected before 
time t=0: curve Co.sub.2. As also shown in FIG. 10c, the mid-point M.sub.C 
of the rising edges of the curves Co.sub.1 and Co.sub.2 of the received 
waves does not depend on the SWH value, but is strongly influenced by the 
mean sea level N.sub.moy. This point corresponds to the point P.sub.S of 
specular reflection, i.e. a point on the surface of the earth. The rising 
edge of the curve Co.sub.1 is very low compared to .tau.. 
Altimetry, and in particular ocean altimetry, is carried out in accordance 
with the invention by deducing from the determined propagation delays of 
direct and reflected waves the coordinates of the specular reflection 
point relative to the position of the receiver R on the ellipsoid 
representing the terrestrial sphere. The reference coordinate system is a 
fixed coordinate system centered on a model defining the terrestrial 
sphere, in a preferred embodiment the WGS-84 model. The receiver R to 
terrestrial sphere T vectors derive from measurements carried out when the 
receiver R (a low Earth orbit satellite, for example) moves along its 
orbit are used to construct a curved rigid line which, when the orbit of 
the receiver R has been precisely determined in the same system of 
reference (the WGS-84 model, for example), is used to acquire the height 
of the surface of the seam (or, more generally, of the terrestrial 
surface) relative to the reference (WGS-84) ellipsoid. 
There are basically two sources of errors in the determination of the 
relative coordinates between the receiver R and the point of specular 
reflection. The first, shown in FIG. 11a, is due to the uncertainty d.tau. 
in the measurement of the delay corresponding to the half power point of 
the rising edge of the received waveform (FIG. 10c). This gives a possible 
range of values dh of altitude along the normal to the terrestrial 
ellipsoid and between two iso-range ellipsoids +d.tau./2 and -d.tau./2 for 
the point P.sub.S of specular reflection. 
The second source of error, shown in FIG. 11b, is due to the so-called 
"normal deviation". The normal deviation at a point P.sub.S on the actual 
terrestrial surface T is the angular difference .alpha. between the normal 
n' to this surface and the normal n to the reference (WGS-84) ellipsoid. 
Because of this difference, which is assumed to be unknown, there is an 
uncertainty ds in the location of the point P.sub.S of specular 
reflection. However, as shown below, the contribution of the normal 
deviation to the vertical error is negligible. 
Consider the case of a "GPS" satellite again, by way of non-limiting 
example. Table 5 in APPENDIX 3 of this description sets out the values of 
the pulse limited footprint dimensions as a function of the elevation 
angle .beta. for the two types of codes previously mentioned: "C/A" and 
"P", and the arc values. The footprint has an elliptical shape with its 
greatest dimension in the plane of incidence. The dimensions have been 
calculated for a receiver R on board a low Earth orbit ("LEO") satellite 
altitude 700 km. 
Referring now to FIG. 12, the point P.sub.S of specular reflection can be 
located on the surface of the terrestrial sphere by three coordinates: the 
relative time-delay .tau. between the reflected signal and the direct 
signal and the two angles .alpha..sub.1 and .alpha..sub.2 defining the 
normal deviation. In FIG. 2 only one of the two angles .alpha., that in 
the plane of incidence, is shown, for simplicity. The first coordinate 
.tau. is measured by the altimeter and fixes and ellipsoid l.sub.iso in 
space. The .alpha. coordinates are given by the topography of the surface 
of the terrestrial sphere, for example the surface of the ocean in the 
intended application. They define a single point at which the ellipsoid 
l.sub.iso and the surface of the earth are tangential. This is the 
characteristic associated with the point P.sub.S of specular reflection. 
The altitude of the terrestrial surface relative to the reference (WGS-84) 
ellipsoid at the point of specular reflection can therefore be written as 
a function of .tau., .alpha..sub.1 and .alpha..sub.2 : 
##EQU18## 
In the following description it is assumed that the altimeter processes the 
direct and reflected signals in such a way that the delay corresponding to 
the point P.sub.S of specular reflection is measured with the precision 
given by d.tau.. The point P.sub.S of specular reflection is a point on 
the terrestrial sphere corresponding to the mean sea level and the 
associated delay is determined by the half power point of the rising edge 
of the received signal (FIG. 10c). 
The angles .alpha. depend directly on the shape of the terrestrial sphere 
relative to the reference (WGS-84) ellipsoid. The value of the normal 
deviation over the ocean is very small, in the order of d.alpha.=10.sup.-5 
radians or even less. For this reason its effect on the vertical accuracy 
can be ignored. 
The partial derivatives of equation (56) above are calculated next. For the 
calculation of 
##EQU19## 
see FIG. 13. The following equation can be written directly: 
##EQU20## 
where the .theta. is the angle of incidence and c is the speed of light. 
For the computation of 
##EQU21## 
see FIG. 14. The following equations can be deduced from this figure: 
By combining equations (55 ) through 57): 
##EQU22## 
Because the very small angles .alpha. are squared in this expression, their 
contribution can be ignored and therefore the vertical accuracy can 
finally be approximated by the formula: 
##EQU23## 
For calculating numerical values from this equation it is necessary to know 
the precision of the time-delay measurement inherent to the altimeter. For 
this purpose it is possible to extrapolate the precision obtained by the 
NASA altimeter for the Topex/Poseidon project. This altimeter is capable 
of a precision of 4 cm in measuring time-delays using a 320 MHz bandwidth 
pulsed chirp signal. 
If a transmitter of the "GPS" system is used, for "P" (precision) type 
codes with a bandwidth of 10 MHz, and allowing for the ratio of the 
bandwidths, the achievable precision is: 
##EQU24## 
For example, the vertical precision of the altimeter can be calculated for 
a receiver R on a satellite in low earth orbit ("LEO"): altitude 700 km, 
as a function of the elevation angle of the transmitter ("GPS") satellite. 
These values are set out in table 6 in APPENDIX 3 at the end of this 
description. 
As mentioned already, a plurality of signal transmitters E.sub.1 through 
E.sub.n can by used. In this case there is a different point P.sub.S of 
specular reflection for each transmitter above the local horizon of the 
receiver R, as shown in FIG. 15. In general, the power available from the 
signals transmitted is relatively low and it follows that the down-looking 
antenna of the receiver R must have good directivity to achieve the 
required "signal to noise ratio". The size of the antenna is critical and 
depends directly on the power level of the signals used. 
The normalized monostatic radar cross-section .sigma..sub.0, .rho..sub.b 
=.rho..sub.0 (0) at the terrestrial surface T (ocean surface) is large at 
nadir but decreases rapidly with increasing angle of incidence. The rate 
of change of this cross-section depends also on the significant wave 
height parameter previously defined and the wind speed. However, as 
already mentioned, the instrument operates in pulse limited mode for which 
the footprint is so small that variations in the radar cross-section with 
angle of incidence can be ignored. Applying bistatic radar theory. It has 
already been shown that the normalized bistatic radar cross-section in the 
case of specular reflection is exactly the same as the normalized 
monostatic radar cross-section at nadir: .rho..sub.b =.rho..sub.0 (0). As 
shown in FIG. 15, every pulse limited signal footprint, regardless of the 
transmitter E.sub.1 through E.sub.n, has a large normalized cross-section 
for the receiver R. 
Depending on the directivity of the antenna of the receiver R, the antenna 
footprint may enclose only one footprint of the pulse limited signal and 
the beam then has to be pointed to each specular reflection point in turn, 
as shown in FIG. 15. In this case the antenna of the receiver R is 
advantageously of the "phased array" so that the beam can be swept over an 
area of the ocean surface sufficient to enclose all points of specular 
reflection. At a given time the cone of the beam intercepts an area A of 
beam enclosing the area .alpha. of the footprint of a pulse limited 
signal. 
In the case of a receiver R on a low Earth orbit ("LEO") satellite at 700 
km altitude and transmitters of "GPS" system satellites, all specular 
reflection points at an elevation angle exceeding 40.degree. correspond to 
off-nadir angles less than 40.degree.. In this case the antenna of the 
receiver R must be designed so that the direction of the receive diagram, 
which points down, can be steered within a cone with a half-angle of 
40.degree. and its axis parallel to nadir. 
The receiver R must also be provided with an up-pointing second antenna to 
receive the signals direct from the transmitter(s). These signals are used 
as reference signals in processing of the reflected signals. They provide 
the response which would be measured by the down-looking antenna from the 
target point. For this reason the reflected signals are processed by 
correlating them with the direct signal. 
The altimeter device for implementing the method of the invention and the 
more specific features of the method are now described with reference to 
FIGS. 16 through 18. 
FIG. 16 shows an altimeter device of the invention. The direct signal 
S.sub.D and reflected signal S.sub.R enter at respective inputs E.sub.D 
and E.sub.R. The transmitters are assumed to transmit signals modulated 
with a pseudo-random code, which is actually the case for transmitters of 
the "GPS" system satellites. The device of the invention includes a known 
type signal processor 1. This can calculate the foreseeable Doppler 
frequency of the reflected signal from the positions and velocities of the 
receiver R and the transmitted E (one specific transmitter if several 
transmitters are used, as in FIG. 15) received at a particular time. It is 
assumed that these parameters (position and speed) are either known or can 
be calculated conventionally. In the conventional way, the direct signal 
S.sub.D is first transposed to as lower frequency by a fixed frequency 
first control down converter 10, under the control of the signal processor 
1 which supplies a first frequency f.sub.1. The transposed signal is then 
shifted in frequency using the Doppler frequency f.sub.D calculated by the 
signal processor 1 to obtain the same frequency shift as that affecting 
the reflected signal S.sub.R. The frequency of the latter is also 
converted by a down converter 12 driven by the aforementioned frequency 
f.sub.1 . As mentioned already, the signal processor 1 also determines an 
approximate value of the relative delay between the direct signal S.sub.D 
and the reflected signal S.sub.R from the point of specular reflection. 
The direct signal S.sub.D after its frequency has been down converted and 
shifted is then delayed by this amount (13), with the exception n/2 "code 
chips", where n is the number of segments defining the delay variation 
within which the mean sea level is estimated. This estimate is obtained 
with reference to the chosen terrestrial model, for example the "WGS-84" 
model in the preferred embodiment of the invention. This replica of the 
direct signal is then passed through a discrete delay line 2 with n output 
to a series of first inputs of a bank 3 of n correlators 3.sub.1 through 
3.sub.n. The second inputs receive the reflected signal, the frequency of 
which has been down converted by the converter 12 but which has not 
undergone any other processing. Each element 2.sub.1 through 2.sub.n of 
the discrete delay line 2 delays the transmitted signal by the same amount 
.tau., with the result that the signal at the output of the nth correlator 
3.sub.n is delayed by n.rho.. Each correlator 3.sub.1 through 3.sub.n 
correlates a delayed replica of the direct signal with the undelayed 
reflected signal. The delay is between .rho. and n.rho., depending on the 
stage in question of the discrete delay line 2. As there are n 
correlators, connected to n outputs of the delay line, correlation to n 
different time positions is obtained. The output of each correlator is 
detected by the detectors 4.sub.1 through 4.sub.n of the detector circuits 
4 to generate samples of the received wave power, as shown in FIG. 10c. 
Each of these samples can be interpreted as a quantity of power reflected 
by the ocean surface at a given range relative to the direct signal. 
The reflected samples may be contaminated by thermal noise and speckle, 
with speckle being the dominant source of noise if the system has been 
properly designed with respect to the "signal to noise ratio". Speckle can 
be caused by the different rate of change of the phase of the components 
of the reflected signal corresponding to different portions of ocean 
surface. Assuming a random distribution of the phase of the different 
components and following the central limit theorem, the amplitude of the 
output signal of each correlator will have a gaussian distribution. The 
distribution of the power of the output signal will be a negative 
exponential distribution. The mean value of the power will therefore be 
the same as its standard deviation, and time averaging over a number of 
samples will be necessary to improve the estimation of the mean power. A 
rough estimation of the averaging time is 10% of the pulse limited 
footprint, which leads normally to a number of samples greater than 50, 
although a larger time interval could also be used to ensure independence 
between samples. 
In the previously mentioned scenario of a receiver R on board a low earth 
orbit ("LEO") satellite at 700 km altitude and one or more "GPS" system 
satellite transmitters, 10 % of the pulse limited footprint corresponds to 
about 1.5 km or 0.2 seconds at orbital speed, which is 143 times the 
coherence time. In theory this many independent samples could be averaged. 
The following considerations have to be taken account of, however. 
The standard deviation .rho..sub..tau.ave is given by the equation: 
##EQU25## 
where B is the bandwidth, M.sub.s is the number of samples being averaged 
and .rho..tau.=2/B is the sample time resolution. In order to reach an 
accuracy of 128 cm, as previously mentioned, a minimum of 550 samples 
would be required, which corresponds to 0.77 seconds, 5.8 km on the ocean 
surface or 38% of the pulse limited footprint. 
FIG. 17 is a diagram in which the upper part shows the power received 
(reflected signal) as a function of the delay and the lower part shows the 
amplitude of the signals at the output of the detectors 4.sub.1 through 
4.sub.n also as a function of the delay. Knowing the propagation speed of 
radio waves, the delays can be expressed directly in units of length. The 
curve in the upper part of FIG. 17 is entirely analogous to the curve 
shown in FIG. 10c. The curve in the lower part has the advantage of being 
in the form of steps corresponding to the signals S.sub.1 and S.sub.n. It 
is therefore a simple matter to discriminate the mid-power center "step" 
from the other "steps" on either side of it: S.sub.1 through S.sub.3 
"floor" steps and S.sub.5 through S.sub.7 "ceiling" steps. This determines 
the position of the point P.sub.S of specular reflection. 
In addition to the circuits just described with reference to FIG. 16, in a 
preferred embodiment of the invention two main feedback loops are used. 
The first fixes the power difference between the noise affecting the lower 
part of the received signal and its linear upper part. The other maintains 
the received power at the same relative position to the time resolution 
cells. 
The first loop is implemented using one sample N.sub.1 of the header of the 
received signal which conveys very little of the wanted signal power, 
being mainly noise, and various samples S.sub.1 -S.sub.7 distributed 
around the half-power point of the rising edge of the received signal. The 
second loop is implemented using the samples Ec.sub.1 and Ec.sub.2 around 
the central sample Ec.sub.C (half-power). 
FIG. 18 illustrates this feature in more detail. The outputs s.sub.1 
through s.sub.7 of the stages of the correlator bank 3 are connected to 
the inputs e.sub.1 through e.sub.7 of a first adder 5 which also applies 
weighting by 7. The outputs s.sub.3 through s.sub.5 are connected to the 
inputs of a second adder 6 forming the samples E.sub.1, E.sub.C and 
E.sub.2 previously mentioned. It also applies weighting by 3. The outputs 
of the adders 5 and 6 are connected to first inputs of two subtractors 7 
and 8, respectively. These receive at a second input, forming the sample 
N.sub.1 previously mentioned, the output signal s.sub.1. As already 
mentioned, this signal consists essentially of the noise component. The 
output of the subtractor 8 represents the half power of the received 
signal. It is passed to the inverting inputs of respective operational 
amplifiers 9 and 9'. The non-inverting input of the operational amplifier 
9' is connected to a constant reference signal. The AGC output signal of 
the amplifier 9' is a feedback signal for the first feedback loop and acts 
as an automatic gain control signal. The non-inverting input of the 
amplifier 9 is connected to the output of the subtractor 7. The output 
signal SEP, providing a tracking error signal, is passed to the second 
feedback loop. 
The AGC signal controls a variable gain amplifier 14 operating on the 
reflected signal. The signal SEP introduces a variable delay into the 
direct signal by means of the variable delay circuit 13. 
In addition to the two feedback loops the device includes conventional 
altimeter tracker circuits 15 implemented in the signal processor 1 which 
adjusts the delay to be applied (also by operating on the circuits 13, as 
already mentioned) to the direct signal before correlating it with the 
reflected signal so that the middle sample has the same mean power in the 
range between the noise and the flat upper part of the samples. 
The delay adjustment constitutes the measurement of the mean sea level 
while the precision of the altimeter is given by the tracking error 
signal. 
The measurements can be processed subsequently on board the platform in 
ways which are not within the scope of the invention. 
The number of samples per signal naturally depends on the bandwidth of the 
system and also on the data rate of the instruments. To give a numerical, 
but non-limiting, example, and returning to the scenario as previously 
explained: "GPS" satellite transmitters and receiver R on board a low 
Earth orbit ("LEO") satellite, a wave can contain seven samples and the 
data rate is then 5,000 samples per second for the transmitter. 
The invention is naturally not limited to the embodiments specifically 
described with reference to FIGS. 10a through 18 in particular. 
Specifically, altimetry is not restricted to maritime cartography (seas 
and oceans). 
As mentioned, a plurality of different on board receivers can be used. 
Also, specular reflection is not the only phenomenon which can be used. 
Diffuse reflection can be used in a similar way, in particular for 
altimetry of sea ice. 
TABLE 1 
______________________________________ 
APPENDIX 1 
h = 700 km 
Elevation .beta. (deg) 
Arc length S (kg) 
Distance R.sub.2 (km) 
______________________________________ 
0 1551 1774 
20 936 1209 
40 557 913 
60 299 768 
80 95 707 
90 0 700 
______________________________________ 
TABLE 2 
______________________________________ 
h = 10 km 
Elevation .beta. (deg) 
Arc length S (km) 
Distance R.sub.2 (km) 
______________________________________ 
0 204.8 205.2 
20 26.9 28.7 
40 11.8 15.5 
60 5.7 11.5 
80 1.7 10.1 
90 0 10 
______________________________________ 
TABLE 3 
______________________________________ 
h = 1 km 
Elevation .beta. (deg) 
Arc length S (km) 
Distance R.sub.2 (km) 
______________________________________ 
0 65.052 65.064 
20 2.746 2.923 
40 1.197 1.56 
60 0.577 1.155 
80 0.182 1.016 
90 0 1 
______________________________________ 
TABLE 4 
__________________________________________________________________________ 
APPENDIX 2 
Parameter 
C/A P Remarks 
__________________________________________________________________________ 
P.sub.e G.sub.e 
28 dBw 25 dBw "GPS" minimum +3 dB 
R.sub.e 
24 .times. 10.sup.6 m 
24 .times. 10.sup.6 m 
.theta. 
36 deg 36 deg Edge of swath 
K.sub.B 
1.38 .times. 10.sup.-23 W/kHz 
1.38 .times. 10.sup.-23 W/kHz 
T.sub.a 
290 K 290 K 
B 2 MHz 20 MHz Radio frequency bandwidth 
.lambda. 
0.19 m 0.19 m 
F 2 dB 2 dB 
.alpha. 
27 .times. 54 m.sup.2 
8.5 .times. 17 m.sup.2 
"GPS" elevation &gt;40.degree. 
G.sub.r 
37 dB 37 dB 4 .times. 4 m antenna 
A 43 .times. 53 km.sup.2 
43 .times. 53 km.sup.2 
Edge of swath 
.sigma..sub.0 (0) 
12 dB 12 dB ERS-1 altimeter 
SNR.sub.0 
-9 dB -32 dB Single pulse 
R.sub.2 
913 km 913 km Edge of swath 
.rho..sub..alpha. 
27 km 8.5 km "GPS" elevation &gt;40.degree. 
v 7.5 km/s 7.5 km/s Satellite altitude: 700 km 
T.sub.C 
428 .mu.s 1.36 ms 
N 428 13 600 
SNR.sub.N 
17.3 dB 9.3 dB N pulses 
__________________________________________________________________________ 
TABLE 5 
______________________________________ 
APPENDIX 3 
Elevation .beta. 
Arc length to 
Pulse footprint length 
(degrees) P.sub.S (km) C/A (k/m) P (km) 
______________________________________ 
0 1 551 42 
10 1 206 32 
20 936 25 
30 726 20 
40 557 54 17 
50 418 46 15 
60 299 42 13 
70 193 39 12 
80 95 37 12 
90 0 36 11 
______________________________________ 
TABLE 6 
______________________________________ 
Elevation Arc length to 
Vertical precision 
(degrees) P.sub.S (km) 
(cm) 
______________________________________ 
0 1 551 231 
10 1 206 162 
20 936 124 
30 726 101 
40 557 87 
50 418 77 
60 299 71 
70 193 67 
80 95 65 
90 0 64 
______________________________________