A sequential post-selection testing procedure for univariate congruence models with nested and non-nested hypotheses

Abstract

Least-squares-based testing procedures for unstable-point identification in geodetic monitoring networks are vulnerable to the smear effect, whereby the influence of a true displacement spreads over several coordinate differences. This leads to a displaced point classified as stable (masking) and a stable point classified as unstable (swamping), a problem that becomes more severe when several points move simultaneously. Recent sequential and combinatorial procedures reduce these effects, but they often lack explicit control of the stepwise false-alarm rate and do not treat post-selection in a formal way. This paper presents SEQCUP, a sequential combinatorial post-selection testing procedure for univariate congruence models when the number and location of displaced points are unknown. The method uses two-epoch observation differences, remains invariant with respect to datum definition, and retains a strictly linear congruence model. At each stage, SEQCUP compares the current null model with higher-dimensional alternatives by means of a quadratic-form statistic built from the difference between their orthogonal projection matrices. The critical value is calibrated with Monte Carlo simulations under the parameterized null displacement model, conditional on the data-driven model selected at the previous stage, so that the resulting test remains valid for both nested and non-nested hypotheses within a unified framework. A stopping rule also limits the maximum number of points inspected in the sequential procedure. It relies on the network topology, excludes models that share the same projector, and uses a normalized distance between projectors to avoid stages with potentially weak separability and pronounced smear effects. Numerical experiments with trilateration, GNSS baseline, and levelling networks, together with literature-based scenarios, show that SEQCUP controls false alarms effectively and attains high mean success rates for model identification over a wide range of signal-to-noise ratios. The method performs at least as well as classical procedures and remains comparable to contemporary combinatorial and information-criterion-based methods, with clear advantages in several scenarios involving multiple displaced points and low-to-moderate signal-to-noise ratios.