Observables of the Global Positioning System are related to a nominal time delay
r i =tR -t between receiver clock reading t R =tR,,, +atR, at the epoch of signal arrival and satellite clock reading t S =t&, +6tS , at the epoch of signal departure. The observed delay can be related to true delay and satellite-to-receiver distance p i
through the known velocity of light c , since
where x S =x (tZue ) is the satellite and x R = x R (t,,,, ) the receiver position at the corresponding epochs of signal departure and arrival.
In code observations the delay r; is related to pseudo-distances p i =czi , and the model for the "observables" becomes
where inertial satellite coordinates x S are usually assumed to be known and con- verted to terrestrial ones u s =RxS through a known earth rotation matrix R . Thus the model unknowns are the receiver terrestrial coordinates uR and the satellite and receiver clock errors 6t and hR , respectively.
When phases are observed, however, the observable is not related to the delay r i , but to the remainder @; of its division with the period T =- f 1 , where f is the fre- quency of the signal. Thus 7: = N ~ T +@; =A+@; NS f where N ; is an unknown integer, and the model becomes
Multiplication with the velocity of light c and introduction of the signal wavelength 1=cT =r , leads to the model for phase pseudo-distances L: =em;
f
where the ionospheric I; and the tropospheric T l influence of atmospheric refrac- tion has been included. The only unknowns of interest are the receiver coordinates u, , the rest being "nuisance parameters". Clock errors are eliminated by taking sin- gle and double differences
The atmospheric influences are computed or eliminated, and the remaining un- knowns are either station coordinates, which have real values and the integer number of cycles N ; or their linear combinations, e.g.
The presence of integer unknowns is the new feature, which makes the classical least squares solution not applicable, because it is designed specifically for real un- knowns. The solution of the least squares problem #=(b- Ax) * P(b -Ax) =min by
*=0, ax is possible only for a real-valued variable x , since the derivative with respect to an integer variable makes no sense. We are therefore faced with a new least squares problem of the form
b=Ax+v=[A, A,] [:I +v=A,z+A,n+v, #=vTQ-lv=min ,
with real unknowns z and integer ones n . Due to the discrete nature of n , the prob- lem must be solved in two steps. In the first step, we take separately any value of n , we keep it fixed and we find the corresponding optimal Z =Z(n) which minimizes 4 ,
Fig. 4: The geometry of the least-squares optimal solution when the model b=y + v , y = A, z + A ,n includes both real ( z ) and integer unknowns ( n ).
To any fixed integer-valued n corresponds an optimal value y = y(n) = A, 2 + A ,n
of the observables, which is closest to the data b among all values y = A, z + Ann
and a corresponding optimal value of the real unknowns 2 = %(n) . As n varies among all integer-valued vectors, there exists one, denoted by n I , giving the observable value
f I =y(nl ) closest to b among all y(n) , which together with the corresponding real values 2 I = z(nI ) constitutes the desired optimal solution. The least-squares solution obtained by pretending that n is real-valued (float solution) corresponds to observables
9 = A, 2 + Ann closest to the data b , with corresponding "optimal" real values of the unknowns fi and i . The Pythagorean theorem llb-y112=llb-f112 +llf-y112, where Ilb -fll=const. , allows to replace the minimization of Ilb -yl12 with the mini- mization of II f - 7 112= (n - n)T Q;' (n - n) , by varying n over all integer-valued vectors to obtain the optimal solution nI , 2 I and f I = A, 2 I + A ,fi I .
In the second step we let n vary (in a discrete way!) until $(Z(n)) is minimized in order to obtain the final optimal values 2, and n, , where the subscript I empha- sizes the fact that n, has integer values. Thus the second step is
The solution to the first step is a classical least-squares solution for the modified model b-Ann=Azz+v ( n is known and fixed), i.e.
To this "estimate" corresponds a value of the observables
and of the minimized quantity
What remains is to minimize &(n) by examining all possible integer values of n .
This seems to be an impossible task without some external guidance which will limit the search to a smaller finite set of values of n . The required help comes from the usual least-squares solution i , n , obtained by pretending that n is real valued (floating solution)
where
It can be shown that the corresponding estimates of the observables ?=A?=
=A, f + Ann satisfy the orthogonality relation
which allows us to set the quantity to be minimized in the form
Since Ilb -f1I2 =const. , we can minimize instead
It can be shown (see e.g. Dermanis, 1999, for details) that
$'(n)=(n-n)T Q i l (n-n) . (6.32)
In this way one of different search strategies (Leick, 1995, Hofmann-Wellenhof et al, 1979, Teunissen and Kleusberg, 1998) can be incorporated for locating, in a neighborhood of the real-valued (floating) solution n , the optimal value n, satisfy- ing
We conclude by bringing attention to another critical difference between real- and integer-valued unknown estimation, which is in fact the "heart" of GPS data analysis strategies.
In the case of real-valued unknowns assume that an initial set of observations b1 is used in association with the model
for the determination of estimates ib1 and illbl . Let us also assume that the data bl are sufficient to determine 2, with very good accuracy. Next a second set of obser- vations
becomes available which is used (in combination to the previous ones b1 ) to produce updated estimates ]ilb1,,, and illbl,b2 . Due to the original accuracy of the estimates
klbl , the "improvement" ?llbl,b2 -kllbl will be very small but not zero. The esti- mates i, will "slide" to a nearby best value i ,b2 .
On the contrary, in the case where the parameters xl are integer-valued, the integer estimates i l b , are not allowed to "slide" but only to "jump" to a nearby integer esti- mateiilbl,b2. As demonstrated in eq. (6.33), "clo~eness'~ is associated with estimate accuracy (covariance factor matrix Qi ). If i,,,, has been determined with sufficient accuracy, all integer values around it will be too far away and no "jump" to a better estimate is possible and thus ii, ,,* = ii, . This means that once integer unknowns are efficiently determined from a subset of relevant observations, they cannot be fur- ther improved by "small" variations due to the "influence" of the other observations, as in the case of real-valued unknowns.
Therefore, the following strategy is followed for the analysis of GPS data:
Every data set b =A i z +Ahni + v i , (i=1,2,. . .) is separately analyzed for the esti- mation n i of its own particular integer unknowns. These values are then held fixed and the resulting models bi =bi -Ahhi =A i, z+ v i are combined for the optimal es- timation i of the remaining real-valued unknowns, i.e. the coordinates of points oc- cupied by receivers for observations.