+The BNC PPP ambiguity resolution combines two classical methods: the +LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) algorithm for the integer search, +and BIE (Best Integer Equivariant) weighting for the final estimate. +The following describes the individual steps as implemented in src/PPP/ambres.cpp and src/PPP/lambda.cpp. +
+ +Step 1 – Float ambiguity extraction
+The Kalman filter state vector contains all estimated parameters (receiver position, receiver clock,
+troposphere, and carrier-phase ambiguities). The AR module extracts only the ambiguity entries
+into a reduced vector using a design matrix, together with the corresponding sub-block of the
+variance–covariance matrix.
+
Step 2 – Resolvability filtering
+Before entering the search, each ambiguity must pass the following quality gates:
+
Step 3 – Reference ambiguity selection
+One ambiguity per constellation group is chosen as a reference (the one that minimises the sum of
+double-difference variances). It is tightly constrained to its nearest integer via a Kalman
+pseudo-observation update. This step effectively converts zero-difference to single-difference
+ambiguity space and ensures a stable basis for the search.
+
Step 4 – LAMBDA decorrelation
+The ambiguity covariance matrix is decomposed as L·D·LT.
+An integer-preserving transformation matrix Z (with det(Z) = ±1) is
+applied to minimise the correlation between ambiguities. After this step the search is far more
+efficient because the transformed ambiguities are nearly uncorrelated.
+
Step 5 – BIE candidate search
+A sequential search-and-shrink algorithm (SSEARCH) finds the top 100 integer candidate vectors
+ranked by their squared Mahalanobis distance χ². For each candidate i an
+exponential weight is computed:
+
w(i) = exp(−0.5 · (χ²i − χ²best))
++The BIE estimate is then the weighted average over all candidates: +
+aBIE = ∑ w(i) · a(i) / ∑ w(i)
++This is the key difference from plain LAMBDA/ILS, which picks only the single best integer vector. +BIE produces a real-valued weighted combination and is the minimum mean-square error estimator +under a Gaussian distribution. +
+ +Step 6 – BIE variance
+Rather than adopting the optimistic variance of the single best candidate, BIE computes a
+conservative variance that reflects the probability mass spread across all candidates:
+
σ²BIE(j) = ∑ w(i) · (aBIE(j) − a(j,i))²
+ +Step 7 – Fixability decision and constraint imposition
+An ambiguity is considered fixable if both of the following criteria are met:
+
+For all fixable ambiguities, Kalman equality constraints with a tight weight are applied to force +the filter state vector to adopt the integer values. Subsequent filter updates treat these fixed +ambiguities as pseudo-observations until a cycle-slip triggers a reset. +
+ +Comparison with other PPP ambiguity resolution methods
+| Method | +Integer choice | +Output | +
| Rounding | +Nearest integer per ambiguity, independently | +Hard fix, simple but fragile | +
| Bootstrapping | +Sequential conditional rounding | +Hard fix, faster than ILS | +
| ILS / LAMBDA | +Single globally optimal integer vector | +Hard fix, optimal under Gaussian noise | +
| BIE (BNC) | +Weighted combination of top-N candidates | +Soft/weighted fix, minimum MSE estimator | +