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Covariance-Weighted Spectral Delay Fusion With a One-Dimensional Affine Model for High-Precision Distributed Optical-Fiber Sensing

Zhiyang Xue, Huan Huang, Ziang Chen, Zhongxing Tian, Zeyu Feng, Yuhan Jiang, Dongdong Zou, Jun Li, Gangxiang Shen, Yi Cai

arXiv:2609.16791v1eess.SP

TL;DR

Periodic disturbances can produce ambiguous delay estimates, complicating reliable localization. The method fuses correlated wrapped and unwrapped delays in a one-dimensional affine model to estimate position and integer ambiguities jointly. Across the reported groups, it achieves meter-scale repeatability and lower standard deviation than the single-pair methods.

  • Problem

    Periodic phase cycles can support competing position estimates, and a precise delay estimate alone does not establish the correct ambiguity branch.

    Source: §I · p. 2

  • Method

    A covariance-weighted common-position fit combines correlated wrapped and unwrapped delays to estimate disturbance position and integer ambiguities under a one-dimensional affine model.

    Source: B. Joint Localization and Ambiguity… · p. 8, §V · p. 12

  • Results

    The proposed method’s sample standard deviations are 1.007–1.685 m at 500 mV and 0.449–1.324 m at 1 V, lowest in all reported groups.

    Source: §IV · p. 11

  • Takeaways & Limitations

    The common-position fit tests delay consistency because deviations confined to some pairs generally cannot be explained by a single position displacement.

    Source: §V · p. 12

    OTDR references provide only coarse positional information and cannot verify meter-scale or submeter absolute localization accuracy.

    Source: §IV · p. 9

Abstract

from arXiv · show

Periodic disturbances can produce ambiguous delay estimates, limiting reliable high-precision localization in distributed optical-fiber sensing. We develop spectral delay fusion for a sensing system using a dual-wavelength bidirectional Mach-Zehnder interferometer, with four phase traces recovered by heterodyne detection and digital demodulation. With calibrated propagation parameters and timing offsets fixed, the six pairwise delay predictions form a one-dimensional affine line segment parameterized by the position of a single dominant disturbance, with sensitivities determined by propagation direction and chromatic dispersion. A generalized least-squares estimator combines unwrapped delays from robust cross-spectral phase slopes with wrapped delays from polarity-invariant phase alignment to jointly estimate position and integer ambiguities under the proposed model, using an effective joint covariance to account for shared-channel and cross-representation dependence. Experiments use a 131.335-km sensing fiber at 1530 and 1550 nm, with periodic phase perturbations applied at five nominal positions from 25 to 125 km. Across the reported groups of 20 records, the proposed method yields sample standard deviations of 1.007-1.685 m at a drive voltage of 500 mV and 0.449-1.324 m at 1 V. The ratio of the smallest single-pair sample standard deviation to that of the proposed method ranges from 2.57 to 19.56 at 500 mV and from 2.12 to 2900 at 1 V. The upper ratio reflects unstable single-pair phase-slope delay estimates for periodic disturbances in the 1-V, nominal 50-km group, where the proposed covariance-weighted fusion maintains meter-scale localization repeatability.

I. INTRODUCTION

Periodic disturbances can leave multiple plausible position estimates even when an individual delay is precise. The paper combines complementary delay measurements in a calibrated dual-wavelength, bidirectional MZI model and reports meter-scale repeatability across its tested positions and drive levels.

I. INTRODUCTION

  • A precise local delay does not ensure the correct position: periodic phase cycles can support competing candidates, and correlated pairwise observations require covariance-aware fusion.Distinct positions can remain difficult to distinguish when wrapped-observation separations are small relative to effective delay errors.Source: §I · p. 2
  • The proposed estimator fits unwrapped cross-spectral phase-slope delays and wrapped polarity-invariant phase-alignment delays jointly, estimating position and integer ambiguities with their effective joint covariance.The position is solved for each candidate ambiguity vector, leaving a coupled integer least-squares problem.Source: §I · pp. 2–3

A DUAL-WAVELENGTH BIDIRECTIONAL MZI

  • Both wavelengths probe the sensing fiber in opposite directions, yielding four phase traces and six pairwise delays with different position sensitivities.With calibrated model coefficients fixed, the delay predictions form a one-dimensional affine line segment parameterized by disturbance position.Source: §I · p. 2

A. System Model

  • The system model derives the recovered phase traces from heterodyne detection and digital demodulation, while retaining residual laser phase noise, electrical noise, and waveform distortion.The traces contain channel-dependent amplitude and polarity; the common-component model does not cover arbitrary channel-dependent phase responses.Source: §I · p. 2, A DUAL-WAVELENGTH BIDIRECTIONAL MZI · p. 3, A. System Model · p. 5

I. INTRODUCTION

  • 1.007–1.685 m at 500 mV and 0.449–1.324 m at 1 V are the proposed method’s sample standard deviations, lowest in every reported group.The smallest single-pair standard deviation is 2.57–19.56 times larger at 500 mV and 2.12–2.90 × 10^3 times larger at 1 V.Source: §I · p. 3

B. Pairwise Delay–Position Relations

The six measured delays are modeled as position-dependent physical delays plus relative timing offsets and residual error, with sensitivities shaped by propagation direction and chromatic dispersion.

B. Pairwise Delay–Position Relations

  • The six physical pairwise delays combine propagation delays from two wavelengths and directions, with dispersion setting the group-delay difference per unit length.The measured delay vector adds position-independent relative timing offsets and residual delay error.Source: B. Pairwise Delay–Position Relations · p. 5
  • Relative timing offsets have three independent degrees of freedom and satisfy pairwise closure.The offsets collect channel- and wavelength-dependent timing contributions.Source: B. Pairwise Delay–Position Relations · p. 5
  • Same-direction inter-wavelength inversions require nonzero dispersion sensitivity κ, while the individual-pair inversions provide sensitivity interpretations for the joint model.The delay sign depends on the retained order of the two channels.Source: B. Pairwise Delay–Position Relations · p. 5

2) Same-wavelength counter-propagating delays:

The supplied passage identifies the wavelength index used for this pairwise-delay subsection.

2) Same-wavelength counter-propagating delays:

3) Inter-wavelength counter-propagating delays:

Delay-estimation and calibration errors limit how well pairwise inversions distinguish candidate positions.

3) Inter-wavelength counter-propagating delays:

  • When |κ| ≪ γ1+γ2, same-direction inter-wavelength delay inversions are more weakly conditioned than counter-propagating pairs.Candidate discrimination depends on predicted delay differences relative to estimation and calibration errors.Source: 3) Inter-wavelength… · p. 5

III. SPECTRAL DELAY FUSION WITH A ONE-DIMENSIONAL AFFINE MODEL

Section III formulates a one-dimensional affine model for six pairwise delays and derives unwrapped and wrapped delay estimates from cross-spectra, then uses effective joint covariance to estimate position and integer ambiguities and analyze identifiability and periodic ambiguity.

III-A

  • Section III-A formulates the six pairwise delays as a one-dimensional affine model and derives unwrapped and wrapped delay estimates from cross-spectra.Source: §III · p. 5

III-B

  • Section III-B jointly estimates position and integer ambiguities using the effective joint covariance.Source: §III · p. 5
  • Section III-B analyzes position identifiability and periodic ambiguity.Source: §III · p. 5

A. One-Dimensional Affine Model and Cross-Spectral Delay Estimation

With calibrated propagation parameters and relative timing offsets fixed, the six pairwise delays are modeled as functions of one disturbance position. Cross-spectral phase observations provide the unwrapped and wrapped delay estimates fitted to that model.

One-Dimensional Affine Model

  • A single dominant disturbance is assumed per observation window, with propagation parameters and calibrated relative timing offsets held fixed during localization.The formulation also requires a sufficiently coherent common component over a nonzero frequency span.Source: A. One-Dimensional Affine Model and… · p. 5
  • All six predicted delays form a one-dimensional affine line segment parameterized by the disturbance position, conditional on fixed model coefficients and relative timing offsets.The model leaves position as the search variable rather than treating timing offsets as additional free variables.Source: A. One-Dimensional Affine Model and… · pp. 5–6

Cross-Spectral Delay Estimation

  • Coherence and spectral magnitude indicate phase-observation reliability, while the cross-spectrum model allows relative polarity but not arbitrary remaining frequency-dependent channel phase.The reliability indicators are not evaluated as functions of candidate position.Source: A. One-Dimensional Affine Model and… · p. 6

1) Unwrapped Delay From Phase Slope:

The unwrapped delay is estimated from phase variation across reliable frequency intervals using a robust phase-slope fit. The estimate depends on selecting a delay branch and requires a nonzero frequency span.

1) Unwrapped Delay From Phase Slope

  • The unwrapped delay is estimated from phase slope within connected reliable frequency intervals, with separate intercepts accommodating independent phase-unwrapping constants.The interval-specific intercepts are nuisance parameters, not estimates used to remove arbitrary phase offsets from the wrapped observation.Source: 1) Unwrapped Delay From Phase Slope · p. 6
  • A robust phase-slope estimate uses phase-reliability weights and a robust loss such as the Huber loss.Source: 1) Unwrapped Delay From Phase Slope · p. 6
  • With uniformly spaced frequencies, phase differences identify delay modulo 1/∆f, so the unwrapped delay remains tied to a selected branch within a prescribed interval.Source: 1) Unwrapped Delay From Phase Slope · p. 6
  • At least one connected reliable frequency interval must have nonzero span; a single spectral line cannot determine a slope.Source: 1) Unwrapped Delay From Phase Slope · p. 6

2) Wrapped Delay From Reference-Frequency Phase:

The wrapped delay is obtained by aligning phase observations at a reference frequency after removing a π polarity reversal. It is correlated with the unwrapped observation, and the model does not remove arbitrary channel phase offsets.

2) Wrapped Delay From Reference-Frequency Phase

  • The wrapped delay is formed by aligning phase observations to a common reference frequency and averaging doubled phases to remove unknown real-valued polarity.Source: 2) Wrapped Delay From… · pp. 6–7
  • The wrapped-delay ambiguity period follows from the polarity-invariant phase representation and need not equal a burst-repetition interval.Source: 2) Wrapped Delay From… · p. 7
  • The wrapped-delay model removes a π phase reversal but not an arbitrary channel phase offset, requiring the real-gain common-component assumption.A residual constant phase that is not an integer multiple of π shifts the ideal wrapped delay.Source: 2) Wrapped Delay From… · p. 7
  • Because the wrapped and unwrapped observations use the same phase traces, they are correlated; high phase-alignment magnitude does not establish the correct integer ambiguity.Source: 2) Wrapped Delay From… · p. 7

B. Joint Localization and Ambiguity Analysis

The section formulates joint position and integer-ambiguity estimation from unwrapped and wrapped observations of the same predicted delays. It uses an effective covariance for their dependencies and analyzes when wrapped observations can distinguish competing positions.

Joint estimator

  • Both delay representations estimate the same pairwise delays, whose predictions at a candidate position repeat the affine vector azv + b.For a candidate integer ambiguity vector, the wrapped delays are restored to the unwrapped scale before forming the augmented observation.Source: B. Joint Localization and Ambiguity… · p. 7
  • The effective joint covariance accounts for correlations among pairs sharing phase traces and dependence between unwrapped and wrapped delay errors.The latter dependence includes that introduced when unwrapped delays are used for reference-frequency alignment; the covariance is an effective weighting model unless independently validated as a sampling covariance.Source: B. Joint Localization and Ambiguity… · p. 7
  • For each integer ambiguity vector, the constrained position minimizer is computed analytically, leaving a coupled integer least-squares search rather than independent pairwise rounding.The fitted delays lie on the one-dimensional affine line, and inconsistent cycle indices cannot fit a common position exactly.Source: B. Joint Localization and Ambiguity… · pp. 7–8

Ambiguity analysis

  • Adjacent ambiguity branches can remain difficult to distinguish when wrapped delay differences proportional to κ are small relative to effective delay errors.Exact identifiability does not establish that noisy observations can resolve the remaining separation.Source: B. Joint Localization and Ambiguity… · p. 8
  • Unwrapped phase slopes add position-discriminating information through coherent frequency dependence, provided periodic-disturbance spectra are resolved and phase is sufficiently reliable.This relies on the common-component model and does not add a separate position-dependent score based on correlation magnitude.Source: B. Joint Localization and Ambiguity… · p. 8
  • Positive curvature for a fixed ambiguity vector characterizes conditional position sensitivity, not global ambiguity resolution.Localization repeatability and agreement with an independently known position therefore assess different aspects of performance.Source: B. Joint Localization and Ambiguity… · p. 8

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The experiments evaluate covariance-weighted localization of periodic disturbances using a dual-wavelength, bidirectional interferometer. Across five nominal positions, the proposed method provides meter-scale repeatability, including where single-pair estimates become unstable.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

  • The experiment uses a 131.335-km sensing fiber, 1530- and 1550-nm carriers, and periodic phase perturbations at 500 mV and 1 V.Five nominal positions span 25–125 km; OTDR references provide only coarse positional information and cannot verify meter-scale absolute accuracy.Source: §IV · pp. 8–9
  • Four recovered phase traces yield six pairwise delays, estimated using robust cross-spectral phase slopes and reference-frequency phase alignment.The traces are detrended, bandpass-filtered, and decimated; the six pairs include direction- and wavelength-dependent observations.Source: §IV · p. 9
  • The joint fit uses fixed calibrated propagation and timing parameters, an effective covariance, and an integer search to combine all six delay pairs.The wrapped observations contribute a conditional position-dependent penalty while accounting for their correlation with unwrapped errors.Source: §IV · pp. 9, 10, 12
  • In the illustrated nominal-100-km records, joint-estimate offsets are −1.057 m at 500 mV and +0.154 m at 1 V, each closer to its group mean than the single-pair estimates.The localization-cost plots compare the joint estimate with four single-pair estimates.Source: §IV · p. 10, Figure 4 · p. 10, Figure 6 · p. 10
  • 1.007–1.685 m at 500 mV and 0.449–1.324 m at 1 V: the proposed method has the smallest sample standard deviation in all five groups at both drives.The figures plot standard deviation against mean estimated position; Tables I and II report the group values.Source: §IV · p. 11, Figure 7 · p. 11, Figure 8 · p. 11
  • At 500 mV, similar position sensitivities across single-pair methods mean their uneven repeatability reflects measured delay errors rather than differing delay-to-position gains.The best-performing single pair varies by nominal distance.Source: §IV · p. 11
  • In the 1-V, nominal 50-km group, every single-pair standard deviation exceeds 3.3 × 10^3 m, while fusion yields 1.163 m.Affected pairs have effective unwrapped-delay uncertainty scales above 100 µs; the unwrapped-only fit gives 4.928 m on the same records.Source: §IV · p. 11

V. CONCLUSION

The conclusion presents a spectral-delay fusion method that fits six pairwise delays to a one-dimensional affine position model and jointly estimates disturbance position and integer ambiguities. In the 131.335-km periodic-disturbance experiment, reported groups show meter-scale repeatability, while coarse OTDR references do not establish meter-scale absolute accuracy.

V. CONCLUSION

  • With propagation parameters and timing offsets fixed, the six pairwise delay predictions form a one-dimensional affine segment parameterized by the position of one dominant disturbance.A common-position fit tests whether the delays are mutually consistent; deviations confined to some pairs generally cannot be explained by one position shift, though model-aligned errors can still shift the estimate.Source: §V · p. 12
  • The fusion separates unwrapped-delay fitting from the conditional contribution of restored wrapped delays, using an effective covariance to account for shared-channel and cross-representation dependence.Wrapped delays can add phase sensitivity after their correlation with unwrapped estimates is accounted for.Source: §V · p. 12
  • Across reported 20-record groups, proposed-method sample standard deviations were 1.007–1.685 m at 500 mV and 0.449–1.324 m at 1 V.These statistics quantify localization repeatability, not absolute position error.Source: §V · p. 12
  • Smallest-single-pair/proposed-method standard-deviation ratios were 2.57–19.56 at 500 mV and 2.12–2.90 × 10^3 at 1 V.The upper ratio arose in the nominal 50-km group, where the single-pair standard deviation was 3367.519 m and the joint estimate's was 1.163 m.Source: §V · p. 12
  • Coarse OTDR references cannot verify meter-scale or submeter absolute localization accuracy; further work should address simultaneous disturbances, calibration stability, and more accurate independent position references.Periodic candidates may remain difficult to distinguish when dispersion-induced delay differences are small relative to measurement errors.Source: §V · p. 12

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