Determination of radar threat location from an airborne vehicle

• US 5,999,130 A
• Filed: 04/07/1998
• Issued: 12/07/1999
• Est. Priority Date: 04/07/1998
• Status: Expired due to Fees

• Current Assignee: Trimble Navigation Limited (Trimble Inc.)
• Original Assignee: Trimble Navigation Limited (Trimble Inc.)
• Inventors: Snow, Adrian C., Schipper, John F.
• Primary Examiner(s): Tarcza, Thomas H.
• Assistant Examiner(s): PHAN, DAO LINDA
• Application Number:
  US09/057,018
• Time in Patent Office: 609 Days
• Field of Search: 342/455, 342/457, 701/225, 701/300, 701/301
• US Class Current: 342/457
• CPC Class Codes: G01S 5/04 Position of source determined by a plurality of spaced direction-finders
• CPC Class Codes: G01S 7/021 Auxiliary means for detecting or identifying radar signals or the like, e.g. radar jamming signals

Abstract

A method for estimating the location of the source of a (possibly unfriendly) electromagnetic probe signal (threat radar) received by an aircraft as the aircraft moves along a chosen path. A signal direction finder on the aircraft determines geometric parameters describing a probe signal plane that contains a probe signal propagation direction, as received at the aircraft, at two or more selected times. A location determination system on the aircraft determines the location of the aircraft at each of the selected times. Information on the probe signal plane and the aircraft location is used to estimate initial location coordinates and, optionally, velocity coordinates and acceleration coordinates, for the probe signal source, in two dimensions or in three dimensions. Probe signal planes at additional selected times can be used to provide a check on, or provide a more reliable estimate for, the probe signal source location. The probe signal source may be transported on a land-based, water-based or airborne vehicle.

Claims

1. A method for estimating the location of a source of probe signals of electromagnetic radiation received by an aircraft, the method comprising the steps of:

receiving a probe signal at an aircraft at K distinct selected times (K≧

2), determining one or more selected parameters that describe a probe signal surface that contains the probe signal propagation direction as the probe signal arrives at the aircraft, and determining the location coordinates of the aircraft, at each selected time;

using the probe signal surface parameters and the aircraft location coordinates for the selected times to estimate the location of a source for the probe signal for at least one selected source location time;

choosing said probe signal surface to be probe signal plane containing said probe signal propagation direction;

choosing at least three of said selected parameters the describe the probe signal plane to be proportional to direction cosines of a line extending perpendicularly to the plane; and

choosing said selected number K to be at least 2;

representing a first probe signal plane by a collection of two-dimensional Cartesian coordinates (x, y) that satisfy a first relation

space="preserve" listing-type="equation">(x-x.sub.a,i)cos(φ

.sub.a,i-φ

PS,i)-(y-y.sub.a,i)sin(φ

.sub.a,i-.phi.PS,i)=0(i=1);

representing a second probe signal plane by a collection of two-dimensional Cartesian coordinates (x,y) that satisfy a second relation

space="preserve" listing-type="equation">(x-x.sub.a,i)cos(φ

.sub.a i-φ

PS,i)-(y-y.sub.a,i)sin(φ

.sub.a,i-φ

PS,i)=0(i=2);

where (x_a i,Y_a,i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined (cosφ

_a, i, sinφ

_a, I) is a vector representing an orientation direction for said aircraft at a time said probe signal plane parameters are determined, and (cosφ

_PS,i, sinφ

_PS,i) is a vector representing said probe signal plane, for i=1,2;

obtaining a simultaneous solution (x'"'"',y'"'"') for the location coordinates (x,y) that simultaneously satisfy the first and the second relations; and

interpreting the simultaneous solution (x'"'"',y'"'"') as two-dimensional location coordinates for said probe signal source.

2. The method of claim 1, further comprising the step of choosing said selected source location time to coincide with at least one of said selected probe signal receipt times.

3. The method of claim 1, further comprising the step of estimating location coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

4. The method of claim 1, further comprising the step of determining said source location in two dimensions.

5. The method of claim 4, further comprising the steps of:

choosing said selected number K to be at least equal to 4; and

estimating location coordinates and velocity coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

6. The method of claim 4, further comprising the steps of:

choosing said selected number K to be at least equal to 6; and

estimating location coordinates, velocity coordinates and acceleration coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

7. The method of claim 1, further comprising the steps of:

choosing said number K to be at least equal to 3; and

determining said source location in three dimensions.

8. The method of claim 7, further comprising the steps of:

choosing said selected number K to be at least equal to 6; and

estimating location coordinates and velocity coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

9. The method of claim 7, further comprising the steps of:

choosing said selected number K to be at least equal to 9; and

estimating location coordinates, velocity coordinates and acceleration coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

10. The method of claim 1, further comprising three steps of:

choosing said selected number K to be at least 3;

representing a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x-x.sub.a i)α

.sub.a i+(y-y.sub.a,i)β

a,i+(z-z.sub.a,i)χ

.sub.a,i=0(i=1);

representing a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x,y,z) that satisfy a second relation

space="preserve" listing-type="equation">(x-x.sub.a,i)α

.sub.a,i+(y-y.sub.a i)β

.sub.a,i+(z-z.sub.a,i)χ

.sub.a,i=0(i=2);

representing a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x,y,z) that satisfy a third relation

space="preserve" listing-type="equation">(x-x.sub.a,i)α

.sub.a,i+(y-y.sub.a,i)β

.sub.a,i+(z-z.sub.a i)=0(i=3)

where (x_a, i,y_a,i,z_a,i) are the location coordinates of said aircraft at a time said probe signal planes parameters are determined, α

_a,i,β

_a i and χ

_a,i are direction cosines representing said probe signal plane, for I=1, 2, 3;

obtaining a simulataneour solution (x'"'"', y'"'"', z'"'"') for the location coordinates (x, y, z) that simultaneously satisfy the first, second and third relations; and

interpreting the simulataneous solution (x'"'"', y'"'"', z'"'"') as three-dimensional location coordinates for said probe signal source.

11. The method of claim 1, further comprising three steps of:

choosing said selected number K to be at least 6;

representing the location coordinates (x(t), y(t),z(t)) of said probe signal source as a function of time t by the relations

space="preserve" listing-type="equation">x(t)=x.sub.0 +v.sub.x (t-t.sub.0),

space="preserve" listing-type="equation">y(t)=y.sub.0 +v.sub.y (t-t.sub.0),

space="preserve" listing-type="equation">z(t)=z.sub.0 +v.sub.z (t-t.sub.0),

where (x₀, y₀, z₀) are location coordinates for said probe signal source at a selected time t=t₀, (v_x, v_y, v_z) are velocity coordinates for said probe signal at a selected time t=t₀, and at least one of the coordinates x₀, y₀, z₀, v_x, v_y, v_z is unknown;

representing a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a,i+v.sub.x Δ

t.sub.i,0)α

.sub.a i+(y.sub.0 -y.sub.a,+v.sub.y Δ

t.sub.i,0)β

.sub.a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=1);

representing a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x,y,z) that satisfy a second relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a,i+v.sub.x Δ

t.sub.i,0)α

.sub.a,i+(y.sub.0 -y.sub.a,i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=2);

representing a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a third relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a,i+v.sub.x Δ

t.sub.i,0)α

.sub.a,i+(y.sub.0 -y.sub.a,i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=3);

representing a fourth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fourth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a,i+v.sub.x Δ

t.sub.i,0)α

.sub.a,i+(y.sub.0 -y.sub.a,i+v.sub.y Δ

t.sub.i,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=4);

representing a fifth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fifth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a,i+v.sub.x Δ

t.sub.i,0)α

.sub.a,i+(y.sub.0 -y.sub.a,+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=5);

representing a sixth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a sixth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0(i=6),

where (x_a,i,y_a,i,z_a, i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, χ

_a, i, β

a,i and χ

_a,i are direction cosines representing said probe signal plane, for i=1, 2, 3, 4, 5, 6, and Δ

t_i,0=t_i -t₀ ;

obtaining a simultaneous solution (x'"'"'₀, y'"'"'₀, z'"'"'₀, v'"'"'_x, v'"'"'_y, v'"'"'_z) for the location coordinates and velocity coordinates that simulataneously satisfy the first, second, third, fourth, fifth, and sixth relations; and

interpreting the location coordinates

space="preserve" listing-type="equation">x(t)=x'"'"'.sub.0 +v'"'"'.sub.x (t-t.sub.0),

space="preserve" listing-type="equation">y(t)=y'"'"'.sub.0 +v'"'"'.sub.y (t-t.sub.0),

space="preserve" listing-type="equation">z(t)=z'"'"'.sub.0 +v'"'"'.sub.z (t-t.sub.0),

as three-dimensional location coordinates for said probe signal source for at least one time t.

12. The method of claim 1, further comprising three steps of:

choosing said selected number K to be at least 9;

representing the location coordinates (x(t), y(t), z(t)) of said probe signal source as a function of time t by the relations

space="preserve" listing-type="equation">x(t)=x.sub.0 +v.sub.x (t-t.sub.0)+a.sub.x (t-t0)2/2,

space="preserve" listing-type="equation">y(t)=y.sub.0 +v.sub.y (t-t.sub.0)+a.sub.y (t-t0)2/2,

space="preserve" listing-type="equation">z(t)=z.sub.0 +v.sub.z (t-t.sub.0)+a.sub.z (t-t0)2/2,

where (x₀, y₀, z₀) are location coordinates for said probe signal source at a selected time t=t₀, (v_x, v_y, v_z) are velocity coordinates for said probe signal at the time t=t₀, (a_x, a_y, a_z) are acceleration coordinates for said probe signal at the time t=t₀, and at least one of the coordinates x₀, y₀, z₀, v_x, v_y, a_x, a_y, a_z is unknown;

representing a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=1);

representing a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a second relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=2);

representing a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a third relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=3);

representing a fourth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fourth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=4);

representing a fifth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fifth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=5);

representing a sixth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a sixth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=6);

representing a seventh probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a seventh relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=7);

representing an eighth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy an eighth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=8);

representing a ninth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a ninth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i, 0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i, 0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+

space="preserve" listing-type="equation">(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i, 0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=9);

where (x_a, i, y_a,i, z_a, i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, α

_a,i, β

a,i and χ

_a,i are direction cosines representing said probe signal plane, for i=1, 2, 3, 4, 5, 6, 7, 8, 9, and Δ

t_i, 0=t_i -t₀ ;

obtaining a simultaneous solution (x'"'"'₀, y'"'"'₀, z'"'"'₀, v'"'"'_y, v'"'"'_y, v'"'"'_i, a'"'"'_x, a'"'"'_y, a'"'"'_z) for the location coordinates, velocity coordinates and acceleration coordinates that simultaneously satisfy the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth relations; and

interpreting the location coordinates

space="preserve" listing-type="equation">x'"'"'(t)=x'"'"'.sub.0 +v'"'"'.sub.x (t-t.sub.0)+a'"'"'.sub.x (t-t0)2/2,

space="preserve" listing-type="equation">y'"'"'(t)=y'"'"'.sub.0 +v'"'"'.sub.y (t-t.sub.0)+a'"'"'.sub.x (t-t0)2/2,

space="preserve" listing-type="equation">z'"'"'(t)=z'"'"'.sub.0 +v'"'"'.sub.z (t-t.sub.0)+a'"'"'.sub.x (t-t0)2/2,

as three-dimensional location coordinates for said probe signal source for at least one time t.

13. Apparatus for estimating the location of a source of probe signals of electromagnetic radiation received by an aircraft, the apparatus comprising:

a probe signal receiver, carried on an aircraft, that receives a probe signal at the aircraft at K distinct selected times (K≦

2), that determines and issues one or more geometric parameters that describes a probe signal surface that contains the probe signal propagation direction as the probe signal arrives at the aircraft, and that determines and issues the location coordinates of the aircraft, at each selected time; and

a computer that is programmed to receive and use the probe signal geometric parameters and the aircraft location coordinates for the selected times to estimate the location of a source for the probe signal for at least one selected source location time, said probe signal surface chosen to be a probe signal plane containing said probe signal propagation direction and at least three of said selected parameters the describe the probe signal plane are chosen to be proportional to direction cosines of a line extending perpendicularly to the plane, said number K is at least 2 and said computer;

represents a first probe signal plane by a collection of two-dimensional Cartesian coordinates (x, v) that satisfy a first relation

space="preserve" listing-type="equation">(x-x.sub.a,i)cos(φ

.sub.a,i-φ

PS,i)-(y-y.sub.a, i)sin(φ

.sub.a, i-φ

PS,i)=0 (i=1);

represents a second probe signal plane by a collection of two-dimensional Cartesian coordinates (x, y) that satisfy a second relation

space="preserve" listing-type="equation">(x-x.sub.a,i)cos(φ

.sub.a,i-φ

PS,i)-(y-y.sub.a, i)sin(φ

.sub.a, i-φ

PS,i)=0 (i=2);

where (x_a, i,y_a, i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, (cosφ

_a, i, sinφ

_a, i) is a vector representing an orientation direction for said aircraft at a time said probe signal plane parameters are determined, and (cos φ

PS,i, sin φ

PS,i,) is a vector representing said probe signal plane, for i=1, 2;

obtains a simultaneous solution (x'"'"', y'"'"') for the location coordinates (x, y) that simultaneously satisfy the first and the second relations; and

interprets the simultaneous solution (x'"'"', y'"'"') as two-dimensional location coordinates for said probe signal source.

14. The apparatus of claim 13, wherein said selected source location time is chosen to coincide with at least one of said selected probe signal receipt times.

15. The apparatus of claim 13, wherein said computer estimates location coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

16. The apparatus of claim 13, wherein said computer determines said source location in two dimensions.

17. The apparatus of claim 16, wherein said number K is at least 4 and said computer:

estimates location coordinates and velocity coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

18. The apparatus of claim 16, wherein said selected number K is at least 6 and said computer:

estimates location coordinates, velocity coordinates and acceleration coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

19. The apparatus of claim 13, wherein said number K is at least 3 and said computer:

determines said source location in three dimensions.

20. The apparatus of claim 13, wherein said number K is at least 6 and said computer:

estimates location coordinates and velocity coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

21. The apparatus of claim 19, wherein said number K is at least 9 and said computer:

estimates location coordinates, velocity coordinates and acceleration coordinates for said probe signal source at said selected source location time that lies in a time interval between the earliest and the latest of said selected probe signal receipt times.

22. The apparatus of claim 13, wherein said number K is at least 3 and said computer:

represents a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x-x.sub.a, i)α

.sub.a, i+(y-y.sub.a, i)β

a, i+(z-z.sub.a, i)χ

.sub.a, i=0 (i=1);

represents a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a second relation

space="preserve" listing-type="equation">(x-x.sub.a, i)α

.sub.a, i+(y-y.sub.a, i)β

a, i+(z-z.sub.a, i)χ

.sub.a, i=0 (i=2);

represents a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a third relation

space="preserve" listing-type="equation">(x-x.sub.a, i)α

.sub.a, i+(y-y.sub.a, i)β

a, i+(z-z.sub.a, i)χ

.sub.a, i=0 (i=3);

where (x_a a,i,y_a,i,z_a,i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, α

_a,i, β

_a,i and χ

a,i are direction cosines representing said probe signal surface, for i=1, 2, 3;

obtains a simulataneous solution (x'"'"', y'"'"', z'"'"') for the location coordinates (x, y, z) that simultaneously satisfy the first, second, and third relations; and

interprets the simultaneous solution (x'"'"', y'"'"', z'"'"') as three-dimensional location coordinates for said probe signal source.

23. The apparatus of claim 13, wherein said number K is at least 6 and said computer:

represents the location coordinates (x (t), y (t), z (t)) of said probe signal source as a function of time t by the relations

space="preserve" listing-type="equation">x(t)=x.sub.0 +v.sub.x (t-t.sub.0),

space="preserve" listing-type="equation">y(t)=y.sub.0 +v.sub.y (t-t.sub.0),

space="preserve" listing-type="equation">z(t)=z.sub.0 +v.sub.z (t-t.sub.0),

where (x₀, y₀, z₀) are location coordinates for said probe signal source at a selected time t=t₀, (v_x, v_y, v_z) are velocity coordinates for said probe signal at a selected time t=t₀, and at least one of the coordinates x₀, y₀, z₀, v_x, v_y, v_z is unknown;

represents a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=1);

represents a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a second relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=2);

represents a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a third relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=3);

represents a fourth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fourth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=4);

represents a fifth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fifth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=5);

represents a sixth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a sixth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i

     0)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0)β

a,i+(z.sub.0 -z.sub.a,i+v.sub.z Δ

t.sub.i,0)χ

.sub.a,i=0 (i=6);

where (x_a,i,y_a i, z_a,i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, α

_a, i, β

a,i and χ

_a,i are direction cosines representing said probe signal plane, for i=1, 2, 3, 4, 5, 6, and Δ

t_i, 0=t_i -t₀ ;

obtains a simultaneous solution (x'"'"'₀, y'"'"'₀, z'"'"'₀, v'"'"'_y, v'"'"'_y, v'"'"'_z) for the location coordinates and velocity coordinates that simultaneously satisfy the first, second, third, fourth, fifth, and sixth relations; and

interprets the location coordinates

space="preserve" listing-type="equation">x(t)=x'"'"'.sub.0 +v'"'"'.sub.x (t-t.sub.0),

space="preserve" listing-type="equation">y(t)=y'"'"'.sub.0 +v'"'"'.sub.y (t-t.sub.0),

space="preserve" listing-type="equation">z(t)=z'"'"'.sub.0 +v'"'"'.sub.z (t-t.sub.0),

as three-dimensional location coordinates for said probe signal source for at least one time t.

24. The apparatus of claim 13, wherein said number K is at least 9 and said computer:

represents the location coordinates (x(t), y(t), z(t)) of said probe signal source as a function of time t by the relations

space="preserve" listing-type="equation">x(t)=x.sub.0 +v.sub.x (t-t.sub.0)+a.sub.x (t-t0)2/2,

space="preserve" listing-type="equation">y(t)=y.sub.0 +v.sub.y (t-t.sub.0)+a.sub.y (t-t0)2/2,

space="preserve" listing-type="equation">z(t)=z.sub.0 +v.sub.z (t-t.sub.0)+a.sub.z (t-t0)2/2,

where (x₀, y₀, z₀) are location coordinates for said probe signal source at a selected time t=t₀, (v_x, v_y, v_z) are velocity coordinates for said probe signal at a selected time t=t₀, and at least one of the coordinates x₀, y₀, z₀, v_x, v_y, v_z, a_x, a_y, a_z is unknown;

represents a first probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a first relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=1);

represents a second probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a second relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=2);

represents a third probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a third relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=3);

represents a fourth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fourth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=4);

represents a fifth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a fifth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=5);

represents a sixth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a sixth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=6);

represents a seventh probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a seventh relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=7);

represents a eighth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a eighth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+

space="preserve" listing-type="equation">a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=8);

represents a ninth probe signal plane by a collection of three-dimensional Cartesian coordinates (x, y, z) that satisfy a ninth relation

space="preserve" listing-type="equation">(x.sub.0 -x.sub.a, i+v.sub.x Δ

t.sub.i,0+a.sub.x (Δ

t.sub.i,0)2/2)α

.sub.a, i+(y.sub.0 -y.sub.a, i+v.sub.y Δ

t.sub.i,0+a.sub.y (Δ

t.sub.i,0)2/2)β

a,i+(z.sub.0 -z.sub.a, i+v.sub.z Δ

space="preserve" listing-type="equation">t.sub.i,0+a.sub.z (Δ

t.sub.i,0)2/2)χ

.sub.a, i=0 (i=9);

where (x_a, i, y_a, i, z_a, i) are the location coordinates of said aircraft at a time said probe signal plane parameters are determined, α

_a,i, β

a,i and χ

_a,i are direction cosines representing said probe signal plane, for i=1, 2, 3, 4, 5, 6, 7, 8, 9, and Δ

t_i, 0=t_i -t₀ ;

obtaining a simultaneous solution (x'"'"'₀, y'"'"'₀, z'"'"'₀, v'"'"'_x, v'"'"'_y, v'"'"'_z) for the location coordinates and velocity coordinates that simulataneously satisfy the first, second, third, fourth, fifth, sixth, seventh, eighth and ninth relations; and

interpreting the location coordinates

space="preserve" listing-type="equation">x'"'"'(t)=x'"'"'.sub.0 +v'"'"'.sub.x (t-t0)+a'"'"'.sub.x (t-t.sub.0)2/2,

space="preserve" listing-type="equation">y'"'"'(t)=y'"'"'.sub.0 +v'"'"'.sub.y (t-t0)+a'"'"'.sub.y (t-t.sub.0)2/2,

space="preserve" listing-type="equation">z'"'"'(t)=z'"'"'.sub.0 +v'"'"'.sub.z (t-t0)+a'"'"'.sub.z (t-t.sub.0)2/2,

as three-dimensional location coordinates for said probe signal source for at least one time t.