Recalling the discussions in Sect. 7.2, one has sequential observation equation systems
These two equation systems are un-correlated. The sequential problem can be then solved by accumulating the individual normal equations as discussed in Sect. 7.2:
And VTPV can be calculated by using
If Eq. 7.27 is solvable, then the least squares solution can be represented as
For convenience, the estimated vector of X by using the first group of observations is denoted by X1 and the quadratic form of the residuals by
as well as 
Using the formula (Cui et al. 1982; Gotthardt 1978)
where A and B are any matrices, C and D are matrices that can be inversed and
the inversion of the accumulated normal matrix can be represented as Q:
where E is an identity matrix. The total term in the parenthesis on the right-hand side of Eq. 7.36 can be interpreted as a modifying factor for Q1 matrix; in other words, due to the sequential Eq. 7.28, the Q matrix can be computed by multiplying a factor to the Q1 matrix. So sequential least squares solution of Eqs. 7.27 and 7.28 can be obtained:
Mathematically, the solutions of the sequential problem of Eqs. 7.27 and 7.28 that are solved by using accumulation of the least squares method as discussed in Sect. 7.2.1 or using sequential adjustment as discussed above shall be the same. However, in practice, accuracy of the computation is always limited by the effective digits of the computer being used. Such a limit on the effective digits causes an inaccuracy of numerical computation. And this inaccuracy will be accumulated and propagated in further computing processes. By comparing the results obtained with the above-mentioned methods, it is noticed that the sequential method will give a drift in the results. The drift increases with time and is generally not negligible after a long time interval.
