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Geometric Analysis of Doppler-Based Navigation with Low Earth Orbit Satellites

Carlos Caravaca Gallego, Pini Gurfil, Hector Rotstein

arXiv:2609.11296v1cs.ITmath.IT

TL;DR

Doppler-based LEO navigation can estimate position, velocity, clock bias, and clock drift, but its geometry-dependent Jacobian makes volume-based satellite selection a poor guide to GDOP. The paper derives a geometric Jacobian parameterization and an exact Schur-complement inflation formula. It finds that clock-bias coupling cannot be eliminated, while altitude diversity offers partial, limited relief.

  • Problem

    Volume-based satellite selection, useful in pseudorange GDOP, performs poorly for Doppler GDOP, whose eight-state Jacobian has geometry-dependent clock-bias sensitivity.

    Source: §1 · p. 2

  • Method

    The paper parameterizes the Doppler Jacobian geometrically and uses a Schur-complement decomposition to relate GDOP inflation to clock-bias collinearity with the seven-state subspace.

    Source: §1 · p. 2, §5.2 · p. 13

  • Results

    The geometric coupling that makes clock bias observable also produces strictly positive GDOP inflation, while altitude diversity reduces collinearity only partially.

    Source: §8 · p. 19

  • Takeaways & Limitations

    Altitude diversity separates equal-elevation satellites onto distinct sensitivity bands, but available altitude spread provides limited relief and heading is a weaker lever.

    Source: §8 · p. 19

    The closed-form sensitivity result is exact within the simplified model, which neglects Earth-rotation receiver velocity and orbital eccentricity at a few-percent level.

    Source: §3.1.3 · p. 8, §4.2 · p. 11

Abstract

from arXiv · show

The increasing vulnerability of Global Navigation Satellite Systems has motivated renewed interest in Doppler-based navigation using low Earth orbit satellites, which can determine position, velocity, clock bias, and clock drift from carrier Doppler measurements. However, the geometric dilution of precision (GDOP) in this eight-state problem behaves fundamentally differently from the GDOP in pseudorange-based navigation. In particular, volume-based satellite selection, effective for pseudorange-based GDOP minimization, has been empirically found to perform poorly for Doppler GDOP. This paper establishes a geometric foundation for Doppler GDOP characterization. A closed-form geometric parameterization of the Doppler measurement Jacobian in terms of elevation, azimuth, inclination, and altitude ratio is derived. It is shown that the clock bias sensitivity depends on elevation, altitude, and the angle between the satellite velocity vector and the line of sight. This sensitivity produces a correlation with the clock drift that no geometric arrangement can remove. A Schur complement decomposition of the eight-state information matrix is performed, yielding an exact GDOP inflation formula, governed by a collinearity coefficient that measures the alignment between the clock bias column and a seven-state subspace. It is proven that the geometric coupling that renders clock bias observable is the same coupling that inflates estimation covariance, and that satellite altitude diversity reduces the collinearity by separating satellites at equal elevation onto distinct sensitivity bands. An example employing a two-shell satellite configuration illustrates the analysis.

1 INTRODUCTION

Doppler-only LEO navigation estimates all eight position, velocity, and clock states, but its geometry-dependent clock-bias column makes pseudorange-style volume selection unreliable. The paper analyzes that column and its coupling to the other states, deriving an exact GDOP-inflation result and showing how altitude diversity offers only partial relief.

1 INTRODUCTION

  • LEO Doppler navigation can estimate three position components, clock bias, three velocity components, and clock drift from carrier Doppler alone.The introduction notes prior simulations achieving 1–5 m position errors with large LEO constellations.Source: §1 · pp. 1–2
  • Selecting a subset of visible satellites matters because processing every visible LEO signal may be computationally prohibitive, and pseudorange volume heuristics rely on a geometry-independent clock-bias column.Doppler GDOP depends on the structure of its eight-state Jacobian, unlike that pseudorange assumption.Source: §1 · p. 2
  • Moore et al. (2024) found LOS-vector volume uncorrelated with Doppler GDOP (r = −0.18), while full-Jacobian determinant metrics correlated strongly (r = −0.91).Their empirical analysis attributed volume selection’s failure to its omission of velocity diversity; this paper supplies the missing clock-bias analysis.Source: §1 · p. 2
  • The clock-bias column couples to position through shared LOS-rate dependence and to vertical velocity through shared elevation, making its GDOP cost structurally unavoidable.A Schur-complement decomposition expresses the inflation exactly through the column’s collinearity with the seven-state subspace.Source: §1 · p. 2
  • Altitude diversity reduces collinearity by placing equal-elevation satellites on distinct sensitivity bands, while satellite heading provides a weaker reduction.The sensitivity–decorrelation tradeoff limits both levers, and higher-altitude satellites carry weaker Doppler signatures.Source: §1 · p. 2

2 BACKGROUND

The background sets out Doppler-only estimation of eight navigation states and the Jacobian used to analyze its geometry. It contrasts the geometry-independent pseudorange clock-bias column with Doppler’s geometry-dependent sensitivity, which links clock bias to other states.

2 BACKGROUND

  • Eight or more simultaneously visible LEO satellites can determine all eight navigation unknowns from carrier Doppler measurements without pseudorange.The model uses the receiver-to-satellite LOS convention and treats the receiver as stationary on a non-rotating Earth for its simplified formulation.Source: §2.1 · p. 3

2.1 Doppler Measurement Model Jacobian

  • The Doppler Jacobian has position, clock-bias, velocity, and clock-drift blocks; scaling parameters nondimensionalize it, while local ENU variables support geometric analysis.The circular-orbit assumption simplifies the analysis; lower orbital altitude corresponds to stronger Doppler signatures in the model.Source: §2.1 · pp. 3–4

2.2 Pseudorange Geometric Dilution of Precision

  • Pseudorange GDOP is commonly approximated by maximizing the Gram-matrix determinant, which for four satellites corresponds to LOS-tetrahedron volume, but the proxy weakens as satellite count grows.Its basis is a clock-bias column independent of satellite geometry, a property absent from the Doppler problem.Source: §2.2 · p. 4

2.3 The Clock Bias Column

  • Doppler clock-bias sensitivity arises because a clock error shifts the satellite-state evaluation time, changing LOS geometry as the satellite moves.Its shared LOS-rate dependence with position sensitivity makes position and clock-bias errors difficult to distinguish from one Doppler measurement.Source: §2.3 · pp. 4–5

3 GEOMETRIC PARAMETERIZATION OF THE DOPPLER JACOBIAN

The section expresses the Doppler Jacobian in geometric parameters and derives the clock-bias sensitivity and its bounds. It shows that this sensitivity couples clock bias to other state columns, with a sensitivity band that is independent of receiver latitude.

3 GEOMETRIC PARAMETERIZATION OF THE DOPPLER JACOBIAN

  • All Doppler Jacobian elements are expressed in closed form using satellite elevation, azimuth, inclination, and orbital altitude ratio.The parameterization enables direct geometric analysis of the measurement model.Source: §3.1 · p. 5, §3 · p. 5

3.1 Closed-Form Jacobian Elements

  • The Jacobian position block scales with the LOS rate across the sky, while its velocity block depends only on the LOS direction.At lower elevations the LOS-rate magnitude decreases, reducing per-measurement position information.Source: §3.1 · pp. 6–7

3.1.1 First projection – relative acceleration onto the LOS

  • The first clock-bias projection is relative acceleration onto the LOS, with satellite acceleration modeled as centripetal.The clock-bias derivative combines this projection with relative velocity projected onto the LOS rate.Source: §3.1 · p. 7, §3.1.1 · p. 7

3.1.2 Second projection – relative velocity onto the LOS rate

  • The second clock-bias projection is relative velocity onto the LOS rate, whose geometry depends on the velocity–LOS aspect angle.That aspect angle measures the LOS orientation relative to the orbital plane.Source: §3.1 · pp. 6–7, §3.1.2 · p. 7

3.1.3 Clock-bias sensitivity

  • Clock-bias sensitivity is nonpositive above the horizon and is bounded by in-plane and cross-track envelopes at fixed elevation and altitude.Its aspect-angle-dependent component lies in the span of the position columns; the per-satellite magnitude is at most 1 in that satellite’s normalization.Source: §3.1.3 · p. 8

3.2 Geometric Invariances

  • The sensitivity band depends only on elevation and altitude ratio, not receiver latitude, satellite azimuth, or orbital inclination.Receiver latitude can constrain the aspect angle and thus the realized value within the band, but the coupling is intrinsic to Doppler measurement geometry.Source: §3.2 · pp. 8–9

4 CLOCK-BIAS COUPLING AND THE OBSERVABILITY–CORRELATION TRADEOFF

The section isolates clock-bias cost by comparing seven- and eight-state formulations, then characterizes its sensitivity and couplings. It shows that the same LOS-rate coupling that makes clock bias observable also inflates covariance, with some correlations unavoidable.

4.1 The Seven-State and Eight-State Formulations

  • Separating H7 from H8 by appending the clock-bias column isolates its covariance cost through the bordered Gram matrix and Schur complement.The seven-state formulation contains position, velocity, and clock drift; H8 restores clock bias as its final column.Source: §4.1 · p. 9, §4 · p. 9

4.2 Structure of the Clock-Bias Coupling

  • Term 2 scales with squared LOS-rate, aligning clock bias with position columns, while drift and vertical-velocity couplings remain sign-definite but horizontal couplings may cancel.Symmetric constellations can make the horizontal velocity couplings vanish.Source: §4.2 · pp. 10–11
  • For every above-horizon satellite, c < 0 and |c| ≤ 1, reaching −1 only at zenith; at the horizon it can vanish only for an in-plane line of sight.At the horizon, sensitivity varies with the aspect angle between the line of sight and satellite velocity.Source: §4.2 · p. 10
  • Under the stationary, non-rotating-Earth model, per-satellite sensitivity is strictly negative above the horizon; percent-level receiver motion can lift near-floor entries to zero or above.The aggregate result depends on the negativity of the sum, which a near-zero entry does not affect.Source: §4.2 · p. 11

4.3 The Observability–Correlation Tradeoff

  • The shared LOS-rate dependence makes clock bias observable but necessarily aligns its column with the seven-state subspace, inflating covariance.The resulting covariance inflation is set by geometry and quantified through the Schur complement.Source: §4.3 · p. 11

5 SCHUR COMPLEMENT BOUNDS

The Schur complement isolates the clock-bias column’s contribution to eight-state covariance, with its collinearity against the seven-state subspace governing GDOP inflation. Because perfect decorrelation is impossible, the section characterizes the geometric limits and tradeoffs.

5.1 Schur Complement and Covariance Inflation

  • Adding clock bias yields a positive-semidefinite rank-one covariance increment, so no original state’s covariance decreases and GDOP rises.The eight-state GDOP is strictly greater than the seven-state GDOP when the eight-state Jacobian has full column rank.Source: §5.1 · p. 12

5.2 Geometric Interpretation and the Collinearity Coefficient

  • The Schur complement is the squared norm of c orthogonal to col(H7), making the residual projection the geometric source of inflation.The collinearity coefficient measures the fraction of c within the seven-state column space.Source: §5.2 · pp. 12–13

5.3 Bounds and Limiting Cases

  • For any above-horizon satellite ζ2 > 0, while identical elevation, altitude, and aspect angle yield ζ2 = 1 and unbounded eight-state GDOP.The latter configuration makes the clock-bias column a scalar multiple of the clock-drift column.Source: §5.3 · p. 13
  • GDOP trades total sensitivity d against collinearity ζ2; when ζ2 < 1/2, seven-state geometry provides a bounded proxy for the eight-state problem.This condition means the clock-bias column is more orthogonal than parallel to the seven-state subspace.Source: §5.3 · p. 14
  • At fixed altitude, high-elevation satellites raise d while driving |rc,z| toward unity, so sensitivity and decorrelation cannot rise together.When all satellites share one elevation and aspect angle, the clock-bias and velocity z-columns are proportional.Source: §5.3 · p. 14

6 ALTITUDE DIVERSITY AND THE SENSITIVITY–DECORRELATION TRADEOFF

Altitude diversity separates equal-elevation satellites into distinct clock-bias sensitivity bands, reducing collinearity, while higher altitudes weaken Doppler sensitivity. The section characterizes this tradeoff and the resulting tendency toward extreme altitudes for satellites that carry appreciable coupling.

6.1 Altitude Diversity as a Decorrelation Mechanism

  • At equal elevation, different altitudes produce different clock-bias sensitivities, reducing ζ2 and increasing S; the effect vanishes at zenith apart from scaling.The intrinsic separation between altitudes is greatest for coupling-carrying satellites away from zenith.Source: §6.1 · p. 15, §6.3 · p. 16

6.2 The Sensitivity–Decorrelation Tradeoff

  • Higher-altitude satellites decorrelate c but weaken Doppler sensitivity, so the first high-altitude satellite offers the largest decorrelation benefit and later additions have diminishing returns.The tradeoff is an increase in S only when the decorrelation gain outweighs the sensitivity loss.Source: §6.2 · p. 15

6.3 Implications for Constellation Geometry

  • For coupling-carrying satellites, sensitivity and decorrelation favor altitude extremes, while near-zenith satellites are altitude-degenerate.The split between co-optimal configurations is weakly determined.Source: §6.3 · pp. 15–16
  • Current multi-constellation altitude separation is modest, and its sensitivity cost limits the achievable reduction in ζ2.The Starlink–OneWeb separation cited is Δβ ≈ 0.08.Source: §6.3 · p. 16

7 EXAMPLE: MULTI-ALTITUDE SINGLE EPOCH

A synthetic two-shell example tests how altitude level and altitude diversity affect eight-state Doppler GDOP, then verifies the Schur-complement inflation formula. The optimized geometry places satellites at the altitude extremes, with altitude diversity reducing collinearity and GDOP beyond the improvement from flying lower.

7.1 Setup

  • The example uses a synthetic eight-satellite, 400–1200 km configuration and circular-orbit feasibility constraints for an equatorial receiver.It is an instantaneous Jacobian illustration, not a propagated constellation or design prescription; common reference-altitude normalization preserves relative altitude sensitivity.Source: §7.1 · pp. 16–17

7.2 Numerical Evaluation

  • The optimizer places satellites at the altitude extremes, with seven at 400 km and one at 1200 km in the retained solution.Across 30 restarts, solutions use only the two bounds; seven-plus-one and six-plus-two splits are co-optimal within 1.5% restart variability.Source: §7.2 · pp. 17–18
  • Lowering the single-shell altitude from 780 km to 400 km raises d from 2.87 to 12.17 and lowers GDOP8 from 3.51 to 2.53.Collinearity changes little, from ζ² = 0.918 to 0.911; the Schur complement increases from 0.234 to 1.084.Source: §7.2 · pp. 17–18
  • Raising one satellite from 400 km to 1200 km lowers ζ² from 0.911 to 0.865 and GDOP8 from 2.53 to 2.25.The Schur complement rises from 1.084 to 1.793; the resulting GDOP8 reduction is about 10%.Source: §7.2 · p. 18

7.3 Verification of Theoretical Predictions

  • At the optimized variable-altitude geometry, the predicted GDOP8² − GDOP7² is 1.49, matching the observed difference within 10^-14.The Schur-complement prediction uses (∥ξ∥² + 1)/S; the same section reports a maximum normalized error of 1 × 10^-8 for the closed-form Jacobian rows against finite differences.Source: §7.3 · p. 19

8 DISCUSSION AND CONCLUSIONS

The paper explains Doppler GDOP through the clock-bias column’s geometric coupling to other states and its exact Schur-complement inflation. Altitude diversity can reduce that coupling, but its relief is limited by a sensitivity–decorrelation tradeoff.

8 DISCUSSION AND CONCLUSIONS

  • The same geometric coupling that makes clock bias observable also inflates covariance, with the amount of inflation set by geometry.The Schur complement quantifies the inflation through the collinearity coefficient and total clock-bias sensitivity.Source: §8 · p. 19
  • Altitude diversity reduces collinearity by placing equal-elevation satellites on distinct sensitivity bands, while higher-altitude satellites carry weaker Doppler signatures.In the worked configuration, most improvement comes from flying lower, while diversity itself contributes a smaller gain.Source: §8 · p. 19
  • LOS-vector volume is a poor predictor of Doppler GDOP, whereas a determinant over the full Doppler Jacobian is a strong one.The analysis locates clock-bias sensitivity outside the span represented by LOS volume and concludes that a seven-state proxy is not justified in the examined configurations.Source: §8 · p. 19
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