Ambiguity Function (Cycle Slip Detection and Ambiguity Resolution) (GPS)

It is well-known that in GPS precise positioning, ambiguity resolution is one of the key problems that has to be solved. Some well-derived ambiguity fixing and searching algorithms have been published in the past. One of these methods is the ambiguity function (AF) method, which can be found in many standard publications (Remondi 1984; Wang et al. 1988; Han and Rizos 1995; Hofmann-Wellenhof et al. 1997).

The principle of the ambiguity function method is to use the single-differenced phase observation

tmpD-851

to form an exponential complex function

 

tmpD-852

wheretmpD-853is the phase observable,tmpD-854is the geometric distance of the signal transmitting path, tmpD-855is the wavelength, index j denotes the observed satellite,tmpD-856observational time,


Nis ambiguity,tmpD-857is the model of the receiver clock errors, and i is the imaginary unit. All terms in Eq. 8.48 have the units of cycles and are single-differenced terms. Property

tmpD-863

is used in order to get Eq. 8.50.

Making a summation over all satellites and then taking the modulus operation, one has

tmpD-864

where property

tmpD-865

is used, nj is the satellite number andtmpD-866is the observed satellite number at epoch k. Making a summation of Eq. 8.51 over all the observed time epochs, one has

tmpD-868

wheretmpD-869is the total epochs number. The left side of Eq. 8.52 is called the ambiguity function, where unknowns are the coordinates of the remote station. The values of the ambiguity function have to be computed for all candidates of coordinates, and the optimum solution is found if the function reaches the maximum, i.e.,

tmpD-871

The search area can be determined by the standard deviationstmpD-872of the initial coordinates (e.g., a cube with side lengths oftmpD-873or a sphere with a radius oftmpD-874The AF method is indeed an ambiguity free method. The ambiguity can be computed using the optimal coordinate solution of Eq. 8.53.

Further discussion on the AF method is given in the next sub-section.

Maximum Property of Ambiguity Function

The ambiguity function is discussed in Sect. 8.4. Here a numerical study of the maximum property of the ambiguity function (AF) is given. It seems that the maximum value of the AF trends to be reached at the boundary of any given search area. Numerical examples are given to illustrate the conclusion. However, a theoretical proof has still not been found up to now; even the author tried to find one, but failed.


Numerical Examples

Several numerical examples are given here to illustrate the behaviours of the ambiguity function criterion. The GPS data of the EU AGMASCO project are used. Data are combined with the data of IGS network and solved for precise coordinates as references. The station Faim (N 38.5295°, E 331.3711°) is used as the reference and Flor (N 39.4493°, E 328.8715°) is used as the remote station. The baseline length is about 240 km. The data length is about four hours of 3 October, 1997. KSGsoft is used for computing a static solution of the coordinates of Flor. The differences of the KSGsoft solution and IGS solution are (0.26,1.93,1.37) cm in the global Cartesian coordinate system. Related standard deviations of the KSGsoft solution are (0.04, 0.04, 0.02) cm. The differences are caused partly by the different data lengths. This assures a good standard for the software being used.

The search step is selected as 1 mm. Tropospheric and ionospheric effects are corrected. In the first example, three hours of data are used. The search area is a 3-D cube with side lengths of ±(0.7, 0.7, 0.4) cm in (x,y, z). Results show that the AF maximum is reached at point (-0.7, 0.7, 0.4) cm, which is on the boundary of the area being searched.

A search process (with a search area of ±7 mm and one hour of data) is illustrated in 2-D graphics with the 1st axis containing search numbers and the 2nd axis containing AF values in Fig. 8.4. The graphic looks like a 3-D AF projection of the cubic searching area (the picture could be quite different in other examples). Figure 8.4 clearly shows the boundary maximum effect of the AF criterion. Expanding the searched area (and, of course, its boundary), the maximum is reached on the new boundary (of the new cubic surface).

Alternatively, the search may be made on a spherical surface with an expanding radius. The results of such an example are illustrated in Fig. 8.5, where only radii of 1,2,…, 10 mm are given. As the radius expands, the AF maximum becomes greater and is always reached over the spherical surface with the maximum radius.

3-D coordinate search using ambiguity function

Fig. 8.4. 3-D coordinate search using ambiguity function

Spherical coordinate search using ambiguity function

Fig. 8.5. Spherical coordinate search using ambiguity function

Theoretical Indications

The AF Eq. 8.53 is rewritten as

tmp1E-1_thumb

where Y is the coordinate vector,tmp1E-2_thumbis to be the searched coordinate area and is a closed area (i.e., it includes the boundary r),tmp1E-3_thumbare the residuals of GPS observation equations (a continuous function of Y),tmp1E-4_thumbis a complex function oftmp1E-5_thumbis the modulus of Sk.

If the GPS data sampling intervals are sufficiently close and the numerical integration error is negligible,then one has

tmp1E-10_thumb

wheretmp1E-11_thumbare the beginning and end time of the observations. According to the middle value theorem of the integration (cf., e.g., Bronstain and Semendjajew 1987; Wang et al. 1979), one has a time pointtmp1E-12_thumbso that

tmp1E-15_thumb

i.e., the AF can be represented by a unique G(t) at timetmp1E-16_thumb(the constant factor is omitted here). Equation 8.54 turns out to be

tmp1E-18_thumb

Because of the definition oftmp1E-19_thumbis a modulus of a complex function.

In complex function analysis theory, there is a so-called maximum theorem (cf., e.g., Bronstain and Semendjajew 1987; Wang et al. 1979), i.e.:

Maximum Modulus Theorem: if complex functiontmp1E-20_thumbis analytic within a limited area Z and is continuous over the closed Z, then modulustmp1E-21_thumbreaches the maximum on the boundary r of Z.

However, such a theorem cannot be directly used for Eq. 8.60 because the theorem is valid only for the analytic complex function defined over a complex plane, whereas functiontmp1E-22_thumbis a complicated three-dimensional complex function.

Maybe the interested reader will consider this in detail and find out a theoretical proof.

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