6.4.6 Parametric Adjustment

In this section, we are going to deal with the adjustment of the linear model (6.67), I.E.

AX + С = L (n > u) (6.69)

which, for the adjustment, will be reformulated as:

AX - (L + V) = 0

or

V = АХ - L*) . (6.70)

Here A is called the design matrix, X is the vector of unknown parameters, L is the vector of observations, (L = L* - С where L* is the mean of the observed multisample), and V is the vector of discrepancies, which is also unknown. The formulation (6.70) is known as a set of observation equations.

a

We wish to get such X = X that would minimize the quadratic

T

form V PV in which P is the assumed weight matrix for the observations L (see the previous section). This quadratic form, which is sometimes called the quadratic form of weighted discrepancies, can be rewritten using the observation equations (6.70) as V^TPV = (AX - L)^T P(AX - L)

= ((AX)^T - L^T) (PAX - PL

T T —T T T —T —

= X A PAX - L PAX - X A L + L PL (6.71)

From equation (6.66) we have P = к £_\ where к is a constant scalar and

L

£- is the variance-covariance matrix of L. Since £- is symmetric, the L L

T

weight matrix P is symmetric as well and P = P. We can thus write

L^T = PAX = X^TA^TPL (6.72)

since it is a scalar quantity.

Substituting (6.72) into (6.71) we get

* If we have a non-linear model

L = F(X)

It can be easily linearized by Taylor's series expansion, I.E.

p_v 9F , o_v

(x-x )+...,

x=x

In which we neglect the higher order terms. Putting ax for X-X , al for

L-F(X°) and A ( a matrix) for 9F/3Xi о we get

j x—x

AL = AAX .

This is essentially the same form as equation (6.69). However, in this case we are solving for the corrections AX to the approximate value X of the vector X, instead of solving for X itself.

L = F(X ) ₊ g_x

V^TPV = X^TA^TPAX - 2X^TA^TPL + L^TPL . (6.73)

The quadratic function (6.73), called sometimes the variations function, is to be minimized with respect to X. This is accomplished by equating all the partial derivatives to zero, i.e.

-JL_ _V^TPV =0 i = 1, 2, ... , u, (6.74)

эх¹

and we obtain, writing Ъ/ЪХ for the whole vector of partial derivatives

Э/ЭХ¹,

— V PV = 2X A PA - 2L PA = 0 , ¹)

oX

which can be rewritten as:

^лТ T ~T

X A PA = L PA

or by taking the transpose of both sides we get:

(6.75)

T T -

(A PA)X = A PL

This system of linear equations is called the system of normal equations which can be written, as often used in the literature, in the following abbreviated form:

N X = U (6.76)

T

where N = (A PA) is known as the matrix of coefficients of the normal

T -

equations, or simply the normal equation matrix and U = A PL is the vector of absolute terms of the normal equation.