GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE △P= Observe computed Ax+Δy+一△-+-△τ+v This can be written in matrix form △P= We get such an equation for each satellite in view. In general, for m satellites, we can write this system of m equations in matrix form a ddd AP The equation is often written using matrix symbols as which expresses a linear relationship between the residual observations b(i.e, observed minus computed observations)and the unknown correction to the parameters x. The column matrix v contains all the noise terms, which are also unknown at this point. we call the above matrix equation the "linearised observation equations 4.1.2 The Design Matrix The linear coefficients, contained in the"design matrixA, are actually the partial derivatives of each observation with respect to each parameter, computed using the provisional parameter values. Note that a has the same number of columns as there are parameters, n=4, and has the same number of rows as there are data, m2 4. we can derive the coefficients of a b partial differentiation of the observation equations, producing the following expression
16 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE ∆ ∆ ∆ ∆ ∆τ P P P P x x P y y P z z P v ≡ − = + + + + observed computed ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂τ This can be written in matrix form: ( ) ∆ ∆ ∆ ∆ ∆τ P P x P y P z P x y z = + v ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂τ We get such an equation for each satellite in view. In general, for m satellites, we can write this system of m equations in matrix form: ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆τ P P P P P x P y P z P P x P y P z P P x P y P z P P x P y P z P x y z v v v m m m m m 1 2 3 1 1 1 1 2 2 2 2 3 3 3 3 1 2 3 = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ τ τ τ τ The equation is often written using matrix symbols as: b = Ax + v which expresses a linear relationship between the residual observations b (i.e., observed minus computed observations) and the unknown correction to the parameters x. The column matrix v contains all the noise terms, which are also unknown at this point. We call the above matrix equation the “linearised observation equations”. 4.1.2 The Design Matrix The linear coefficients, contained in the “design matrix” A, are actually the partial derivatives of each observation with respect to each parameter, computed using the provisional parameter values. Note that A has the same number of columns as there are parameters, n = 4 , and has the same number of rows as there are data, m ≥ 4 . We can derive the coefficients of A by partial differentiation of the observation equations, producing the following expression:
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE yo- P Co-x yo-y Note that A is shown to be purely a function of the direction to each of the satellites as observed from the receiver 4.1.3 The least squares solution Let us consider a solution for the linearised observation equations, denoted x. The "estimated residuals" are defined as the difference between the actual observations and the new. estimated model for the observations. Using the linearised form of the observation equations, we can write the estimated residuals as ⅴ=b-Ax The "least squares"solution can be found by varying the value of x until the following functional is minimised J(x)=∑v That is, we are minimising the sum of squares of the estimated residuals. If we vary x by a small amount, then J(x) should also vary, except at the desired solution where it is stationary (since the slope of a function is zero at a minimum point). The following illustrates the application of this method to derive the least squares solution (x)=0 d{b-A)(b-A}=0 (b-A)'(b-A)+(b-Ax)b-Ai)=0 (-Aax)(b-Ax)+(b-Ax)(-A)=0 AAx=Ab The last line is called the system of normal equations. The solution to the normal equations is therefore I=AAAb
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 17 A = − − − − − − − − − − − − x x y y z z c x x y y z z c x x y y z z c x x y y z z c m m m 0 1 0 1 0 1 0 2 0 2 0 2 0 3 0 3 0 3 0 0 0 ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Note that A is shown to be purely a function of the direction to each of the satellites as observed from the receiver. 4.1.3 The Least Squares Solution Let us consider a solution for the linearised observation equations, denoted x . The “estimated residuals” are defined as the difference between the actual observations and the new, estimated model for the observations. Using the linearised form of the observation equations, we can write the estimated residuals as: v = b − Ax The “least squares” solution can be found by varying the value of x until the following functional is minimised: J vi ( ) ( ) i m (x) v v b Ax b Ax T T ≡ = = − − = 2 1 . That is, we are minimising the sum of squares of the estimated residuals. If we vary x by a small amount, then J(x) should also vary, except at the desired solution where it is stationary (since the slope of a function is zero at a minimum point). The following illustrates the application of this method to derive the least squares solution: {( ) ( )} ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) δ δ δ δ δ δ δ δ δ J() x b Ax b Ax b Ax b Ax b Ax b Ax A x b Ax b Ax A x A x b Ax x A b Ax x A b A Ax A Ax A b T T T T T T T T T T T T T = − − = − − + − − = − − + − − = − − = − = − = = 0 0 0 0 2 0 0 0 . The last line is called the system of “normal equations”. The solution to the normal equations is therefore: x (A A) A b T 1 T = −