Utilization of exact solutions of the pseudorange equations

• US 5,914,686 A
• Filed: 08/05/1997
• Issued: 06/22/1999
• Est. Priority Date: 01/11/1997
• Status: Expired due to Term

• Current Assignee: Trimble Navigation Limited (Trimble Inc.)
• Original Assignee: Trimble Navigation Limited (Trimble Inc.)
• Inventors: Schipper, John F.
• Primary Examiner(s): Blum, Theodore M.
• Application Number:
  US08/906,485
• Time in Patent Office: 686 Days
• Field of Search: 342/357, 701/213
• US Class Current: 342/357.25
• CPC Class Codes: G01S 19/42 Determining position
• CPC Class Codes: G01S 19/425 by combining or switching between signals derived from different satellite radio beacon positioning systems

Abstract

A system for determining exact solutions of the pseudorange equations obtained from measurements of location determination (LD) signals received from N LD signal sources by a mobile LD station. The LD signal sources may be part of a GPS, a GLONASS, LEO or any other satellite-based or ground-based LD system, such as Loran. Squares of differences of the pseudorange equations are subtracted from each other to produce N-1 linear relations between the unknown location coordinates x, y and z and the receiver clock offset b. These linear relations are used in one or more squares of pseudorange equations to produce a quadratic equation in b, or in x, y or z. A physically realistic solution b is found and used to determine the remaining unknown coordinates (x,y,z). No iteration or approximation is required. For N≧5, the approach extends to situations where the clock offset varies with time. For N≧5, another approach determines the minimum radius sphere that contains all LD signal antenna locations produced by application of the invention to sets of four LD signals for all permutations of constellations of four LD signal sources. For N≧5, another approach uses mean square error minimization to determine the solution set (x,y,z,b). The invention allows use of less complex receiver clocks and allows simpler and faster signal processing to determine the solution set (x,y,z,b).

Claims

1. A method for determining location fix coordinates for a location determination (LD) station, the method comprising the steps of:

receiving LD signals from a plurality of LD signal sources at an LD station and determining a pseudorange value between the LD station and each LD signal source;

processing the pseudorange values to produce at least one linear relation that exactly relate location fix coordinates for the LD station to location coordinates of at least two of the LD signal sources, with no approximation; and

using the at least one linear relation together with one pseudorange value to produce a nonlinear relation that exactly relates the location fix coordinates for the LD station to the location coordinates of at least one of the LD signal sources, from which the location fix coordinates can be calculated without iteration or approximation.

2. The method of claim 1, wherein said step of processing said pseudorange values to obtain said at least one linear relation comprises the steps of:

measuring a pseudorange value PR(t=t_r,n ;

t_s,n ;

n) associated with an LD signal transmitted by LD signal source number n (n=1, 2, . . . , N), where N is a selected integer at least equal to 2, at a selected time t=t_s,n and received at said LD station at a selected receive time t=t_r,n, where the receive times t_r,n (n=1, 2, . . . , N) are approximately equal, where the pseudorange value for an LD signal received from LD signal source number n at the time t=t_r,n is represented by a pseudorange equation of the form

space="preserve" listing-type="equation">PR(t=t.sub.r,n ;

t.sub.s,n ;

n)+b=d.sub.n ={(x-x.sub.n).sup.2 +(y-y.sub.n).sup.2 +(z-z.sub.n).sup.2 }.sup.1/2 ==b-χ

(t.sub.r,n ;

t.sub.s,n ;

n),

that relates a distance d_n from the LD signal source number n, having known spatial location coordinates (x_n,y_n,z_n)=(x_n (t_s,n),y_n (t_s,n),z_n (t_s,n)) at the selected time t=t_s,n, to said LD station, which has spatial location coordinates (x,y,z), where b/c=Δ

t_r,n is an initially unknown error in the clock coordinate of said LD station, where χ

(t_r,n ;

t_s,n ;

n) is a selected term with a known value that is determined from the measured pseudorange value for a signal transmitted from LD signal source number n and that compensates for at least one of (i) time delay, I_r,s,n, for LD signal propagation in the ionosphere, (ii) time delay, T_r,s,n, for LD signal propagation in the troposphere, and (iii) clock error Δ

t_s,n associated with LD signal source number n, and where x, y, z and b are location fix coordinates, at least one of which is unknown and is to be determined; and

processing the pseudorange equations for the pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce said at least one linear relation relating the location fix coordinates x, y, z and b, where the coefficients of said at least one linear relation are determinable from the selected terms χ

(t_r,n ;

t_s,n ;

n) and are independent of the values of the location fix coordinates x, y, z and b.

3. The method of claim 1, wherein said step of using said linear relation to obtain said nonlinear relation comprises the steps of:

using said at least one linear relation relating said location fix coordinates x, y, z and b, together with at least one of said pseudorange representation equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n), to generate an equation that is nonlinear in said unknown value for a selected one of said location fix coordinates x, y, z and b and that does not depend on the values of said other three location fix coordinates;

determining a selected coordinate value from a solution of the nonlinear equation for said selected location fix coordinate, and using said selected location fix coordinate value in said at least one linear relation relating said location fix coordinates x, y, z and b to determine values of at least one other location fix coordinates that is unknown; and

interpreting the values determined for said location fix coordinates x, y, z and b as the spatial location coordinates and receiver/processor clock error coordinate for said LD signal station at the time t=t_r,n.

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

selecting said integer N to be equal to at least 4;

expressing said at least one linear relation as at least three linear relations relating said solution location fix coordinates in the form

space="preserve" listing-type="equation">H.sub.11 x+H.sub.12 y+H.sub.13 z=A'"'"'.sub.1,2 -B.sub.1,2 b,

space="preserve" listing-type="equation">H.sub.21 x+H.sub.22 y+H.sub.23 z=A'"'"'.sub.1,3 -B.sub.1, 3 b,

space="preserve" listing-type="equation">H.sub.31 x+H.sub.32 y+H.sub.33 z=A'"'"'.sub.1,4 -B.sub.1,4 b,

where the coefficients H_ij (i=1,2,3;

j=1,2,3) are determined by said spatial location coordinates (x_n,y_n,z_n) of said LD signal sources at said selected times t=t_s,n, and the coefficients A'"'"'₁,i (i=2,3,4) and B₁,i (i=2,3,4) are determined by said selected terms χ

(t_r,n ;

t_s,n ;

n).

5. The method of claim 2, wherein said step of processing said pseudorange equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce said at least one linear relation relating said location fix coordinates to said location coordinates of said at least one LD signal source comprises the steps of;

selecting said integer N to be equal to at least 4;

forming three differences of the squares of pairs of said pseudorange equations for four selected LD signal sources n=m and n=ki (m≠

ki;

i=1,2,3;

ki=1, . . . , N), and rearranging each of these differences in the form of a matrix equation ##EQU14## where

space="preserve" listing-type="equation">x.sub.m,k =(x.sub.m +x.sub.k)/2,

space="preserve" listing-type="equation">Δ

x.sub.m,k =(x.sub.m -x.sub.k)/2,

space="preserve" listing-type="equation">y.sub.m,k =(y.sub.m +y.sub.k)/2,

space="preserve" listing-type="equation">Δ

y.sub.m,k =(y.sub.m -y.sub.k)/2,

space="preserve" listing-type="equation">z.sub.m,k =(z.sub.m +z.sub.k)/2,

space="preserve" listing-type="equation">Δ

z.sub.m,k =(z.sub.m -z.sub.k)/2,

space="preserve" listing-type="equation">A'"'"'.sub.m,k =χ

(t.sub.r,m ;

t.sub.s,m ;

m).sup.2 -χ

(t.sub.r,k ;

t.sub.s,k ;

k).sup.2, +x.sub.m,k Δ

x.sub.m,k ++y.sub.m,k Δ

y.sub.m,k +z.sub.m,k Δ

z.sub.m,k,

space="preserve" listing-type="equation">B.sub.m,k =2 χ

(t.sub.r,m ;

t.sub.s,m ;

m)-χ

(t.sub.r,k ;

t.sub.s,k ;

k)!;

determining an inverse matrix H^-1 of the 3×

3 matrix ##EQU15## applying the inverse matrix H^-1 to both sides of the second matrix equation to produce a matrix equation ##EQU16##

6. The method of claim 2, further comprising the steps of:

selecting said integer N to be equal to at least 5;

for at least four distinct sets of said LD signals received from four of said LD signal sources, numbered n=j1, j2, j3, j4 with 1≦

j1<

j2<

j3<

j4≦

N, determining said location coordinates (x'"'"',y'"'"',z'"'"',b'"'"')_j1,j2,j3,j4 of said LD station;

determining location coordinates (x_c,y_c,z_c,b_c) of the center of a sphere in four-dimensional space that has a minimum radius r_min and that contains all points having said location coordinates (x'"'"',y'"'"',z'"'"',b'"'"')_j1,j2,j3,j4 for the at least four distinct sets (j1,j2,j3,j4) of said LD signal sources; and

designating the location coordinates (x_c,y_c,z_c,b_c) as said location fix coordinates of said LD station.

7. The method of claim 2, further comprising the steps of:

selecting said integer N to be equal to at least 5;

processing said pseudorange equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce N-1 linear relations of the form

space="preserve" listing-type="equation">H.sub.k1 x+H.sub.k2 y+H.sub.k3 z=A'"'"'.sub.1,k+1 -B.sub.1,k+1 b (k=1, 2, . . . , N-1),

where the coefficients H_kj (k=1,2, . . . , N-1;

j=1,2,3) are determined by said LD signal source location coordinates, and the coefficients A'"'"'₁,i (i=2, . . . , N) and B₁,i (i=2, . . . , N) are independent of the values of said location fix coordinates x, y, z and b;

forming an error expression ε

of the form ##EQU17## where w₁, w₂, . . . , w_N are selected non-negative constants; and

minimizing the value of the error expression ε

with respect to choices of at least one of the values of said location fix coordinates x, y, z and b.

8. The method of claim 1, wherein said step of processing the pseudorange values to obtain said at least one linear relation comprises the steps of:

measuring a pseudorange value PR(t=t_r,n ;

t_s,n ;

n) associated with an LD signal transmitted by LD signal source number n (n=1, 2, . . . , N), where N is a selected integer at least equal to 3, at a selected time t=t_s,n and received at said LD station at a selected receive time t=t_r,n, where the receive times t_r,n (n=1, 2, . . . , N) are approximately equal, where the pseudorange value for an LD signal received from LD signal source number n at the time t=t_r,n is represented by a pseudorange equation of the form

space="preserve" listing-type="equation">PR(t=t.sub.r,n ;

t.sub.s,n ;

n)+b=d.sub.n ={(x-x.sub.n).sup.2 +(y-y.sub.n).sup.2 +(z-z.sub.n).sup.2 }.sup.1/2 ==b-χ

(t.sub.r,n ;

t.sub.s,n ;

n),

that relates a distance d_n from the LD signal source number n, having known spatial location coordinates (x_n,y_n,z_n)=(x_n (t_s,n),y_n (t_s,n),z_n (t_s,n)) at the selected time t=t_s,n, to said LD station, which has spatial location coordinates (x,y,z) with at least one of the spatial coordinates being known and the remainder being initially unknown, where b/c=Δ

t_r,n is an initially unknown error in the clock coordinate of the LD station, where χ

(t_r,n ;

t_s,n ;

n) is a selected term with a known value that is determined from the measured pseudorange value for the signal transmitted from the LD signal source number n and that compensates for at least one of (i) time delay, I_r,s,n for LD signal propagation in the ionosphere, (ii) time delay, T_r,s,n for LD signal propagation in the troposphere, and (iii) clock error Δ

t_s,n associated with LD signal source number n, and where x, y, z and b are location fix coordinates to be determined; and

processing the pseudorange equations for the pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce at least two linear relations relating the location fix coordinates x, y, z and b, where the coefficients of the at least two linear relations are determinable from the selected terms χ

(t_r,n ;

t_s,n ;

n) and are independent of the values of the location fix coordinates x, y, z and b.

9. The method of claim 8, wherein said step of using said at least one linear relation to obtain said nonlinear relation comprises the steps of:

using said at least one linear relation relating said location fix coordinates x, y, z and b, together with at least one of said pseudorange representation equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n), to generate an equation that is nonlinear in an unknown value for a selected one of said initially unknown location fix coordinates and that does not depend on the values of said other location fix coordinates;

determining a selected coordinate value from a solution of the nonlinear equation for said selected location fix coordinate, and using the selected location fix coordinate value in said at least one linear relation relating said location fix coordinates x, y, z and b to determine values of said other unknown location fix coordinates; and

interpreting the values determined for said location fix coordinates x, y, z and b as the spatial location coordinates and receiver/processor clock error coordinate for said LD signal station at the time t=t_r,n.

10. The method of claim 1, wherein said step of processing the pseudorange values to obtain said at least one linear relation comprises the steps of:

measuring a pseudorange value PR(t=t_r,n ;

t_s,n ;

n) associated with an LD signal transmitted by LD signal source number n (n=1, 2, . . . , N), where N is a selected integer at least equal to 3, at a selected time t=t_s,n and received at said LD station at a selected receive time t=t_r,n, where the receive times t_r,n (n=1, 2, . . . , N) are approximately equal, where the pseudorange value for an LD signal received from LD signal source number n at the time t=t_r,n is represented by a pseudorange equation of the form

space="preserve" listing-type="equation">PR(t=t.sub.r,n ;

t.sub.s,n ;

n)+b=d.sub.n ={(x-x.sub.n).sup.2 +(y-y.sub.n).sup.2 +(z-z.sub.n).sup.2 }.sup.1/2 ==b-χ

(t.sub.r,n ;

t.sub.s,n ;

n),

that relates a distance d_n from the LD signal source number n, having known spatial location coordinates (x_n,y_n,z_n)=(x_n (t_s,n)y_n (t_s,n),z_n (t_s,n)) at the selected time t=t_s,n, to said LD station, which has initially unknown spatial location coordinates (x,y,z), where b/c=Δ

t_r,n is a known error in the clock coordinate of said LD station, where χ

(t_r,n ;

t_s,n ;

n) is a selected term with a known value that is determined from the measured pseudorange value for a signal transmitted from LD signal source number n and that compensates for at least one of (i) time delay, I_r,s,n for LD signal propagation signal in the ionosphere, (ii) time delay, T_r,s,n, for LD signal propagation in the troposphere, and (iii) clock error Δ

t_s,n associated with LD signal source number n, and where x, y, z and b are location fix coordinates to be determined; and

processing the pseudorange equations for the pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce at least two linear relations relating the location fix coordinates x, y, z and b, where the coefficients of the at least two linear relations are determinable from the selected terms χ

(t_r,n ;

t_s,n ;

n) and are independent of the values of the location fix coordinates x, y, z and b.

11. The method of claim 10, wherein said step of using said linear relation to obtain said non-linear relation comprises the steps of:

using said at least two linear relations relating said location fix coordinates x, y, z and b, together with at least one of said pseudorange representation equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n), to generate an equation that is nonlinear in the unknown value for a selected one of said initially unknown location fix coordinates x, y and z that does not depend on the values of said other unknown location fix coordinates;

determining a selected coordinate value from a solution of the nonlinear equation for the selected location fix coordinate, and using the selected location fix coordinate value in said at least two linear relations relating said location fix coordinates x, y, z and b to determine values of said other unknown location fix coordinates; and

interpreting the values determined for said location fix coordinates x, y, z and b as the spatial location coordinates and receiver/processor clock error coordinate for said LD signal station at the time t=t_r,n.

12. The method of claim 1, further comprising the step of selecting a first of said LD signal sources from a first Satellite Positioning System and selecting a second of said LD signal sources from a second Satellite Positioning System, where the first and second Satellite Positioning Systems differ from each other.

13. The method of claim 12, further comprising the step of selecting said first Satellite Positioning System to be a Global Positioning System and selecting said second Satellite Positioning System to be a Global Orbiting Navigational Satellite System.

14. The method of claim 1, further comprising the step of selecting a first of said LD signal sources from a Satellite Positioning System and selecting a second of said LD signal sources from a ground-based location determination system.

15. The method of claim 14, further comprising the step of selecting said Satellite Positioning System from the class consisting of a Global Positioning System and a Global Orbiting Navigational Satellite System.

16. The method of claim 14, further comprising the step of selecting said ground-based location determination system from the class consisting of

a Loran signal system, a Tacan signal system, a Decca signal system, an Omega signal system, a JTIDS Relnav signal system and a Personal Location Reporting Service (PLRS) signal system.

17. The method of claim 1, further comprising the step of selecting said first and said second LD signal sources from a Satellite Positioning System.

18. Apparatus for determining the present location of a location determination station, the apparatus comprising:

a location determination (LD) station that receives LD signals from a plurality of LD signal sources and that determines a pseudorange value between the LD station and each LD signal source;

a microprocessor that communicates with the LD station and that is programmed;

to process the pseudorange values to produce at least one linear relation that exactly relates location fix coordinates for the LD station to location coordinates for at least two of the LD signal sources; and

to use the at least one linear relation together with one pseudorange value to produce a nonlinear relation that exactly relates the location fix coordinates of the LD station to location coordinates of at least one of the LD signal sources, from which the location fix coordinates can be calculated without iteration or approximation.

19. The apparatus of claim 18, wherein said LD station and said microprocessor process said pseudorange values to obtain said at least one linear relation by:

measuring a pseudorange value PR(t=t_r,n ;

t_s,n ;

n) associated with an LD signal transmitted by LD signal source number n (n=1, 2, . . . , N), where N is a selected integer at least equal to 2, at a selected time t=t_s,n and received at said LD station at a selected receive time t=t_r,n, where the receive times t_r,n (n=1, 2, . . . , N) are approximately equal, where the pseudorange value for an LD signal received from LD signal source number n at the time t=t_r,n is represented by a pseudorange equation of the form

space="preserve" listing-type="equation">PR(t=t.sub.r,n ;

t.sub.s,n ;

n)+b=d.sub.n {(x-x.sub.n).sup.2 +(y-y.sub.n).sup.2 +(z-z.sub.n).sup.2 }1/2==b-χ

(t.sub.r,n ;

t.sub.s,n ;

n),

that relates a distance d_n from the LD signal source number n, having known spatial location coordinates (x_n,y_n,z_n)=(x_n (t_s,n),y_n (t_s,n),z_n (t_s,n)) at the selected time t=t_s,n, to said LD station, which has spatial location coordinates (x,y,z), where b/c=Δ

t_r,n is an initially unknown error in the clock coordinate of said LD station, where χ

(t_r,n ;

t_s,n ;

n) is a selected term with a known value that is determined from the measured pseudorange value for a signal transmitted from LD signal source number n and that compensates for at least one of (i) time delay, I_r,s,n, for LD signal propagation in the ionosphere, (ii) time delay, T_r,s,n, for LD signal propagation in the troposphere, and (iii) clock error Δ

t_s,n associated with LD signal source number n, and where x, y, z and b are location fix coordinates, at least one of which is unknown and is to be determined; and

processing the pseudorange equations for the pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce said at least one linear relation relating the location fix coordinates x, y, z and b, where the coefficients of said at least one linear relation are determinable from the selected terms χ

(t_r,n ;

t_s,n ;

n) and are independent of the values of the location fix coordinates x, y, z and b.

20. The apparatus of claim 19, wherein said microprocessor uses said linear relation to obtain said nonlinear relation by:

using said at least one linear relation relating said location fix coordinates x, y, z and b, together with at least one of said pseudorange representation equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n), to generate an equation that is nonlinear in the unknown value for a selected one of said location fix coordinates x, y, z and b and that does not depend on the values of said other three location fix coordinates;

determining a selected coordinate value from a solution of the nonlinear equation for the selected location fix coordinate, and using the selected location fix coordinate value in said at least one linear relation relating said location fix coordinates x, y, z and b to determine values of other three location fix coordinates that are unknown; and

interpreting the values determined for said location fix coordinates x, y, z and b as the spatial location coordinates and receiver/processor clock error coordinate for said LD signal station at the time t=t_r,n.

21. The apparatus of claim 20, said integer N is selected to be equal to at least 4 and said microprocessor is further programmed:

to express said at least one linear relation as at least three linear relations relating said solution location fix coordinates in the form

space="preserve" listing-type="equation">H.sub.11 x+H.sub.12 y+H.sub.13 z=A'"'"'.sub.1,2 -B.sub.1,2 b,

space="preserve" listing-type="equation">H.sub.21 x+H.sub.22 y+H.sub.23 z=A'"'"'.sub.1,3 -B.sub.1,3 b,

space="preserve" listing-type="equation">H.sub.31 x+H.sub.32 y+H.sub.33 z=A'"'"'.sub.1,4 -B.sub.1,4 b,

where the coefficients H_ij (i=1,2,3;

j=1,2,3) are determined by said spatial location coordinates (x_n,y_n,z_n) of said LD signal sources at said selected times t=t_s,n, and the coefficients A'"'"'₁,i (i=2,3,4) and B₁,i (i=2,3,4) are determined by said selected terms χ

(t_r,n ;

t_s,n ;

n).

22. The apparatus of claim 20, wherein said integer N is selected to be equal to at least 4 and said microprocessor is programmed:

to form three differences of the squares of pairs of said pseudorange equations for four selected LD signal sources n=m and n=ki (m≠

ki;

i=1,2,3;

ki=1, . . . , N), and rearranging each of these differences in the form of a matrix equation ##EQU18## where

space="preserve" listing-type="equation">x.sub.m,k =(x.sub.m +x.sub.k)/2,

space="preserve" listing-type="equation">Δ

x.sub.m,k =(x.sub.m -x.sub.k)/2,

space="preserve" listing-type="equation">y.sub.m,k =(y.sub.m +y.sub.k)/2,

space="preserve" listing-type="equation">Δ

y.sub.m,k =(y.sub.m -y.sub.k)/2,

space="preserve" listing-type="equation">z.sub.m,k =(z.sub.m +z.sub.k)/2,

space="preserve" listing-type="equation">Δ

z.sub.m,k =(z.sub.m -z.sub.k)/2,

space="preserve" listing-type="equation">A'"'"'.sub.m,k =χ

(t.sub.r,m ;

t.sub.s,m ;

m).sup.2 -χ

(t.sub.r,k ;

t.sub.s,k ;

k).sup.2, +x.sub.m,k Δ

x.sub.m,k ++y.sub.m,k Δ

y.sub.m,k +z.sub.m,k Δ

z.sub.m,k,

space="preserve" listing-type="equation">B.sub.m,k =2 χ

(t.sub.r,m ;

t.sub.s,m ;

m)-χ

(t.sub.r,k ;

t.sub.s,k ;

k)!;

to determine an inverse matrix H^-1 of the 3×

3 matrix ##EQU19## and to apply the inverse matrix H^-1 to both sides of the second matrix equation to produce a matrix equation ##EQU20##

23. The apparatus of claim 20, wherein said integer N is selected to be equal to at least 5 and said microprocessor is programmed:

for at least four distinct sets of said LD signals received from four of said LD signal sources, numbered n=j1, j2, j3, j4 with 1≦

j1<

j2<

j3<

j4≦

N, to determine said location coordinates (x'"'"',y'"'"',z'"'"',b'"'"')_j1,j2,j3,j4 of said LD station;

to determine location coordinates (x_c,y_c,z_c,b_c) of the center of a sphere in four-dimensional space that has a minimum radius r_min and that contains all points having the location coordinates (x'"'"',y'"'"',z'"'"',b'"'"')_j1,j2,j3,j4 for the at least four distinct sets (j1,j2,j3,j4) of said LD signal sources; and

to designate the location coordinates (x_c,y_c,z_c,b_c) as said location fix coordinates of said LD station.

24. The apparatus of claim 20, wherein said integer N is selected to be equal to at least 5 and said microprocessor is programmed:

to process said pseudorange equations for said pseudorange values PR(t=t_r,n ;

t_s,n ;

n) to produce N-1 linear relations of the form

space="preserve" listing-type="equation">H.sub.k1 x+H.sub.k2 y+H.sub.k3 z=A'"'"'.sub.1,k+1 -B.sub.1,k+1 b (k=1, 2, . . . , N-1),

where the coefficients H_kj (k=1,2, . . . , N-1;

j=1,2,3) are determined by said LD signal source location coordinates, and the coefficients A'"'"'₁,i (i=2, , . . . , N) and B₁,i (i=2, . . . , N) are independent of the values of said location fix coordinates x, y, z and b;

to form an error expression ε

of the form ##EQU21## where w₁, w₂, . . . , w_N are selected non-negative constants; and

to minimize the value of the error expression ε

with respect to choices of at least one of the values of said location fix coordinates x, y, z and b.