8.3.4 Ambiguity Resolution: The Key to Modern GPS Surveying

IMPROVING THE EFFICIENCY OF THE AMBIGUITY SEARCH PROCEDURE

At this point the apriori information may be:

approximate baseline components, with relatively high accuracy.

a range of ambiguity values corresponding to the apriori baseline knowledge.

The more accurate the apriori information, the smaller the search volume (as in Figure 1 below) and the greater the likelihood that the correct position is within the search volume. Furthermore, the smaller the search space, the lower the computational burden.

Figure 1. The search volume as a function of apriori baseline component accuracy.

There are a number of search procedures, each with their own geometric conditions:

• Search volume in the "ambiguity domain", consisting of a hyper-ellipsoid in n-dimensional space (n being the number of double-differenced ambiguities), as used in the Fast Ambiguity Resolution Approach employed in the SKI^TM software (FREI & BEUTLER, 1990). The apriori ambiguities and their uncertainties (determined from a preliminary ambiguity-free solution) define the location, shape and size of the hyper-ellipsoid (see figure 2).
  
• Three-dimensional search volume in the "ambiguity domain", consisting of an ellipsoid or cube within which different sets of "lines-of-ambiguities" intersect. The apriori baseline information and its uncertainty define the location, shape and size of the search volume (Figure in section 8.2.5 is a 2-D example).
  
• Cubic search volume in which the coordinates of the free or "remote" station are expected to lie (the other end of the baseline is the fixed or "monitor" station). The apriori baseline information and its uncertainty define the location and size of the search cube (Figure 1). This method is sometimes referred to as a search technique in the "coordinate domain".

Figure 2. N-dimensional ambiguity search space used for the FARA technique.
(FREI & BEUTLER, 1990)

• Determination of the candidate ambiguity sets directly from the observations, using a combination of carrier phase and pseudo-range measurements. This method is sometimes referred to as a search technique in the "measurement domain".

The Least Squares search procedures for "rapid static" and "on-the-fly" GPS surveying techniques, although similar to those employed in the conventional ambiguity resolution techniques, are a more sophisticated implementation of one or more search and validation procedures.

Ambiguity Domain Techniques

Methods (1) and (2) are two geometric descriptions of "ambiguity domain" search procedures. However, there are quite a number of practical techniques described in the literature (HAN & RIZOS, 1997):

Fast Ambiguity Resolution Approach (FARA) (FREI & BEUTLER, 1990).

Cholesky Decomposition (LANDAU & amp; EULER, 1992).

Spectral Decomposition (ABIDIN, 1993).

Least Squares Ambiguity Search Technique (HATCH, 1990).

Fast Ambiguity Search Filter (FASF) (CHEN, 1993).

Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) (TEUNISSEN, 1994).

 

HAN (1995) discusses the mathematical basis for each of these techniques, and explains how they are all related to each other and to the general theory of Integer Least Squares estimation.

However, there are several factors which affect the geometry of ambiguity intersections and which may be manipulated in order to change the geometry to make the Least Squares search in the 3-D "ambiguity domain" more efficient (that is, fewer candidate ambiguity intersections):

• Order in which the sequential between-satellite differences are made (Figure 3a).
• Whether the between-satellite differencing is sequential, or with respect to a fixed base satellite (Figure in section 7.2.4 and Figure 3b).
• Choice of base satellite, if the fixed base satellite strategy for between-satellite differencing is used (Figure 3c).
• Number of satellites observed (Figure 3d).
• Effective wavelength of the ambiguity cycles (Figure 3e).

Figure 3a. Impact of between-satellite differencing order on candidate ambiguity sets.
(ABIDIN, 1993)

Figure 3b. Impact of between-satellite differencing strategy on candidate ambiguity sets.
(ABIDIN, 1993)

Figure 3c. Impact of selection of base satellite on candidate ambiguity sets.
(ABIDIN, 1993)

Figure 3d. Impact of number of satellites on candidate ambiguity sets.
(ABIDIN, 1993)

Figure 3e. Impact of observation wavelength on candidate ambiguity sets.
(ABIDIN, 1993)

In the above figures, the candidate ambiguity sets are those represented by the open circles, where all "lines-of-ambiguities" intersect at one point (or very close to a single point). The fewer such candidate ambiguity sets within the search volume (either the 3-D space here, or the n-dimensional ambiguity space in the case of the FARA technique), the less testing is necessary to eliminate the bogus ambiguity sets. However, there is still the issue of how obvious is the correct ambiguity set from all possible candidate sets. Very often it is hard to discriminate between the "best" set and the "second best" ambiguity sets when very short observation sessions are processed, hence care is necessary in order to design sensitive testing and rejection criteria that reliably identify only the correct ambiguity set. This is discussed later in the chapter.

Coordinate Domain Techniques

The Ambiguity Function Method (ERICKSON, 1992) is an example of this class of search technique. The AFM has some unique characteristics:

(8.3-2) where m is the number of satellites, j is the number of observation epochs, x,y,z is the coordinate of the trial point, is the observed double-differenced phase (in cycles), and is the calculated double-differenced range based on the trial point coordinates (in cycles).




• Maximum value of AF = (m-1).j
• Create a fine grid over the search cube and test all trial points within the cube, and compare largest to second largest AF value. The 3-D intersection point (trial coordinate) where the maximum value of the AF is found is the final coordinate value for the "remote" station (unknown end of the baseline).
• Ambiguity values are not explicitly obtained, though under ideal conditions the maximum AF value occurs at an intersection of integer "lines-of-ambiguities".
• The method is insensitive to cycle slips, and can be used for static and kinematic applications, with one or more epochs of data.
• The AFM can be applied to any linear combination of L1 and L2 carrier phase observations, and in fact can be applied sequentially in a boot-strapping approach using observables with successively smaller wavelengths (corresponding to the "fineness" of the search grid).
• The AFM technique is computationally intensive (particularly if the search volume is large), and is not well suited for real-time applications.

The AFM is mathematically equivalent to the Least Squares search methods, and hence provides no better discrimination between the "best" and "second best" candidate ambiguity sets (or the coordinates corresponding to the largest and second largest AF values) than other search techniques. One important difference between the AF method and the Least Squares search methods is that apart from the size of the search volume and the resolution or "fineness" of the grid of trial coordinates, the AFM does not take into account any of the geometric conditions that influence the number and distribution of ambiguity intersections.

Measurement Domain Techniques

The basis of this technique is the combination of (double-differenced) carrier phase and pseudo-range data as exemplified by eqn (8.2-1):

(8.3-3)

This relation is valid for either L1 or L2, as well as linear combinations of the two frequencies (such as the "wide-lane" or "narrow-lane" combinations -- section 6.4.4 and section 6.4.5). There are several comments that must be made:

• This is a "geometry-free" or "orbit-free" technique as the geometric range is missing in eqn (8.3-3).
• Using L1 or L2 observations in eqn (8.3-3) is problematic as the ionospheric delay for phase or pseudo-range is opposite in sign, and hence does not cancel in the combination (certain dual-frequency combinations can overcome this).
• The pseudo-range data is quite "noisy", with measurement errors that may be as large as several wavelengths.
• The pseudo-range data is contaminated by multipath errors.

Despite these problems this technique appears to be the simplest and most "direct" of the ambiguity search techniques, and is particularly useful on its own, or in combination with another search procedure. However, it can only be used with top-of-the-line GPS receivers able to measure all four observables (_L1, _L2, P_L1, P_L2).

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© Chris Rizos, SNAP-UNSW, 1999