By the FCB- or IRC-based methods, the integer properties of ambiguities at a single receiver can be retrieved. However, integer resolutions are accomplished through the ambiguity search and validation. Ambiguity search means determining the range of integer candidates for a float ambiguity estimate. Theoretically, this range depends on the float ambiguity estimate and its stochastic statistics. On the other hand, ambiguity validation means deciding whether an integer candidate should be accepted as the correct integer under a given confidence level. The criteria for this validation depend on the presumed statistical distribution of the quadratic form of least-squares residuals (Teunissen and Kleusberg 1998). In the following, the sequential bias rounding by Dong and Bock (1989) and the LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) by Teunissen (1994) for ambiguity resolution is briefly introduced.
2.5.1
Sequential bias rounding
Sequential bias rounding, or integer bias bootstrapping (Teunissen 1998), means that a chosen set of ambiguities are rounded to integers at a particular step, and the estimates and precisions of remaining ambiguities are hence improved for a new bias-rounding step. The rounding procedure is based on the assumption that if a float ambiguity estimate is close to an integer and has a high precision, this integer is highly likely to be the correct one (Dong and Bock 1989). The
2.5 Ambiguity search and validation 33
fixing decision is made according to a probability value, namely
P0= 1 − ∞ X i=1 efrc i − N − Nˆ √ 2σ − efrc i + N − Nˆ √ 2σ (2.41)
where efrc(x) is the complementary error function
efrc(x) = √2π Z ∞
x
e−t2dt (2.42)
and ˆN is the float ambiguity estimate; σ is the precision or the standard deviation of ˆN ; N is the nearest integer to ˆN . For a given confidence level α, an ambiguity can be fixed to its nearest integer if its corresponding P0 is larger than 1 − α, and otherwise not (Dong and Bock 1989; Ge
et al. 2005a, 2008). During the initial stage of sequential bias rounding, there may be only a few ambiguities that can be successfully fixed to integers using the criterion above. Nonetheless, as ambiguity resolution progresses, more ambiguities will be improved and thus can be resolved.
Similarly, Blewitt (1989) proposed a sequential bias optimizing method where the cumulative probability for all correctly resolved ambiguities is calculated and another ambiguity is fixed only if the cumulative probability stays larger than 99%. In addition, based on the above sequential bias bounding, Ge et al. (2006) suggested an integer resolution of a float ambiguity estimate according to only its closeness to an integer without considering its standard deviation. As a result, the ambiguity validation is performed by testing whether the residuals of measurements led by each ambiguity are compatible before and afterwards. Note that this strategy is suggested mainly for speeding up a huge-network analysis.
2.5.2
LAMBDA
The LAMBDA method offers the highest probability of coming up with the correct integer candidates for ambiguities (Joosten and Tiberius 2002). Ambiguity resolution is actually based on the least squares principle (e.g. Hofmann-Wellenhof et al. 2001), namely
S = ˆN − NTQ−1 ˆ
N ˆN − N
= minimum (2.43)
where ˆN is the m-dimensional vector of float ambiguity estimates and N is its corresponding integer vector; QNˆ is the variance-covariance matrix of ˆN. Hence, ambiguity resolution is
achieved by identifying an N that minimizes the sum squares of the ambiguity residual vector ˆ
N − N weighted by the inverse of QNˆ. Due to the integer constraints on the ambiguities,
Equation 2.43 has to be solved by means of a search. However, this search is not an easy task because it is carried out over the whole m-dimensional space of integers. In order to simplify this problem, the whole integer space has to be replaced by a smaller set of integers, which is hereafter called the ambiguity search space. This space is located at ˆN and its shape and orientation are governed by QNˆ. It is in this local space that the search for the integer least-squares solution is
performed (Teunissen 1998).
Teunissen (1994) defined the ambiguity search space as a set of grid points N that satisfies
S = ˆN − N
T
Q−1Nˆ ˆN − N
where the constant χ2 guarantees that the search space contains the correct integer candidate.
For details on how to determine χ2refer to de Jonge and Tiberius (1996). It is worth emphasizing
that the volume of an ambiguity search space is governed by both χ2 and Q ˆ
N. Due to the high
correlation between the float ambiguity estimates in the scenario of a very short observation period, the resulting search space is usually severely elongated and stretched over a considerable range of cycles (Joosten and Tiberius 2002). Consequently, a search over such space is extremely inefficient. Hence, Teunissen (1994) proposed an integer invertible volume-preserving matrix Z to transform the original search space into a sphere-like one, namely
N′
= ZN, ˆN′
= Z ˆN, QNˆ′ = ZQNˆZ
T (2.45)
The search is then performed in this transformed space and the results are finally transformed back (de Jonge and Tiberius 1996). This transformation can be recognized as a decorrelation of ambiguity estimates after which the corresponding variance-covariance matrix QNˆ becomes
more diagonal. Note that Z transforms only the shape and the orientation of the ambiguity search space, but not its volume. In addition, for completeness, latest developments on the LAMBDA method are given in Henkel et al. (2009), Henkel and G¨unther (2010) and Teunissen (2010).
On the other hand, the optimum integers identified by the above LAMBDA method should be validated before being accepted as correct values. As addressed by Teunissen and Kleusberg (1998), ambiguity validation includes an acceptance test and a discrimination test. The accep- tance test can be recognized as the compatibility between the potential integer candidates and the associated GPS measurements (Wang et al. 1998). For example, one possible test is based on the compatibility between the unit variances of the ambiguity-fixed and ambiguity-float solutions (Han 1997b; Teunissen and Kleusberg 1998), namely
σ2 fixed σ2 float < m r − t(Fm,r−t−m;α− 1) + 1 (2.46) where r is the number of measurements; m is the number of ambiguities; t is the number of remaining parameters; Fm,r−t−m;α denotes the F -distribution of a confidence level α with m
and r − t − m degrees of freedom. Furthermore, σ2 fixed σ2 float =r − t − m r − t · Ωfloat+ S Ωfloat (2.47)
where Ωfloat is the quadratic form of least-squares residuals in an ambiguity-float solution. A
smaller σ
2 fixed
σ2 float
indicates a more reliable integer resolution.
Moreover, the discrimination test is to confirm that the optimum integers are statistically significantly better than the second-optimum integers. This test is normally performed by quan- tifying the ratio between the second minimum and the minimum quadratic form of ambiguity residuals, which is hereafter called the ratio test (Euler and Schaffrin 1990), namely
Sopt2
Sopt1
> γ (2.48)
where Sopt2 and Sopt1 correspond to the second-optimum and optimum integers, respectively;