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Cooperative LEO-Terrestrial Multistatic ISAC: CRLB Analysis, Scaling Laws, and Satellite Selection

Yunhui Li, Kaitao Meng, Emad Alsusa, Kaiting You

arXiv:2609.09784v1eess.SP

TL;DR

Down-tilted terrestrial beams can leave aerial targets with weak SCNR and elevation information, motivating cooperative LEO observations for 3-D localization. The paper analyzes PPP and orbit-aware Walker models and uses CRLB-oriented greedy satellite selection; simulations validate the approximations and show performance near exhaustive search at lower complexity.

  • Problem

    Down-tilted terrestrial beams can leave aerial targets with weak SCNR and elevation information, motivating complementary LEO observations for 3-D localization.

    Source: §I · p. 1

  • Method

    The paper analyzes cooperative satellite observations with PPP and Earth-curvature-aware Walker models, then selects satellites greedily using SCNR reliability and geometric complementarity.

    Source: §I · p. 2

  • Results

    Monte Carlo simulations validate both approximations; greedy selection nears exhaustive-search performance at substantially lower complexity, with hybrid-aware selection more beneficial under directionally imbalanced terrestrial deployments.

    Source: §VI · p. 11

  • Takeaways & Limitations

    At fixed satellite density, increasing nearby-satellite density yields faster asymptotic localization gains than extending the cooperation region to include distant satellites.

    Source: §I · p. 2, A. PPP-Based Stochastic Satellite Model · p. 6

Abstract

from arXiv · show

Low Earth orbit (LEO) satellites provide elevated and spatially diverse viewpoints for enhancing three-dimensional (3-D) sensing in integrated satellite-terrestrial networks (ISTNs). This paper investigates a LEO-assisted terrestrial multistatic integrated sensing and communication (ISAC) network for 3-D target localisation, where multiple LEO satellites act as cooperative sensing illuminators and provide additional bistatic observations to distributed terrestrial radar receivers. We first model cooperative satellites as a homogeneous Poisson point process (PPP) and derive a tractable approximation of the average hybrid Cram\'er-Rao lower bound (CRLB). The resulting scaling laws show that the root-CRLB decreases with the inverse square root of the mean number of cooperative satellites for a fixed cooperation region, while increasing the cooperation radius at fixed satellite density yields logarithmic diminishing returns. We then develop an Earth-curvature-aware Walker model incorporating orbital structure, satellite motion, visibility, and time-varying sensing geometry, and derive a tractable approximation of the corresponding hybrid CRLB. Analytical bounds on the marginal gain and a sufficient condition for ordering candidate satellites are obtained. Based on these results, a CRLB-oriented greedy satellite-selection strategy is proposed to account for SCNR-dependent reliability and geometric complementarity with the terrestrial sensing configuration. The proposed strategy consistently outperforms benchmarks and approaches exhaustive-search performance with substantially lower complexity. Monte Carlo simulations validate the analytical approximations for both models.

I. INTRODUCTION

The paper studies cooperative LEO illumination for terrestrial 3-D localization, addressing weak terrestrial sensing geometry for aerial targets. It develops stochastic and orbit-aware models, a hybrid system model, and a CRLB-oriented satellite-selection strategy.

I. INTRODUCTION

  • Terrestrial down-tilted beams can leave aerial targets with weak SCNR and elevation information, while elevated LEO viewpoints provide complementary high-elevation bistatic observations.These observations motivate cooperative LEO sensing for 3-D localization.Source: §I · p. 1
  • The PPP model yields tractable hybrid-CRLB approximations and scaling laws: root-CRLB falls as O(K̄^-1/2) at fixed region, while expanding the region at fixed density gives only logarithmic improvement.Increasing nearby-satellite density provides faster asymptotic localization gains than adding increasingly distant satellites.Source: §I · p. 2
  • The Earth-curvature-aware Walker model captures orbital structure, satellite motion, visibility, and time-varying sensing geometry while supporting a tractable hybrid-CRLB approximation.It bridges local stochastic analysis and orbit-constrained deployments.Source: §I · p. 2
  • The proposed greedy satellite-selection strategy combines SCNR-based reliability with geometric complementarity and approaches exhaustive-search performance at substantially lower complexity.Its marginal-gain analysis provides bounds and a sufficient condition for ordering candidate satellites.Source: §I · p. 2

II. SYSTEM MODEL

  • The system combines a terrestrial base station, distributed radar receivers, LEO satellites, and UAV targets; satellite illumination creates additional bistatic observations at the receivers.The terrestrial and satellite sensing branches share spectrum and use mutually orthogonal waveforms after satellite Doppler and timing compensation.Source: §II · p. 3

A. Channel Model

  • The channel model represents terrestrial and satellite-assisted target and clutter paths through their respective directions and path gains.Satellite-assisted paths run from satellite to target or clutter scatterer and then to a radar receiver.Source: A. Channel Model · p. 3

B. Signal Model

The signal model separates terrestrial communication and sensing transmissions from dedicated satellite sensing signals. Orthogonal satellite waveforms and MVDR receive beamforming support echo detection, with SCNR used to characterize sensing performance.

B. Signal Model

  • The TBS simultaneously transmits communication and sensing signals, with its total power equally allocated among communication and sensing beams.The signal streams and beamforming vectors are defined separately for users and targets.Source: B. Signal Model · p. 3
  • Each LEO satellite transmits dedicated sensing signals toward targets, and its total transmit power is equally allocated among its sensing beams.Satellite probing waveforms are assumed orthogonal, so inter-satellite sensing interference is not considered.Source: B. Signal Model · pp. 3–4
  • Radar receivers use MVDR beamforming to preserve the desired target echo while suppressing interference and noise, and SCNR is the sensing-performance metric.The model includes interference from unintended targets, clutter, and other sensing beams.Source: B. Signal Model · p. 4

III. SENSING PERFORMANCE ANALYSIS

The section analyzes cooperative LEO sensing with a PPP model and derives an approximation to the average hybrid CRLB. Its scaling analysis finds stronger asymptotic gains from adding satellites within a fixed region than from extending that region at fixed density.

A. PPP-Based Stochastic Satellite Model

  • The model places ground projections of cooperative satellites in a homogeneous PPP over a local disk, with Poisson-distributed satellite count and uniform conditional locations.The disk has radius Dmax, density λs, and mean count λsπDmax^2.Source: A. PPP-Based Stochastic Satellite Model · p. 4
  • A Taylor expansion around the mean hybrid FIM yields a tractable PPP-averaged CRLB approximation when random FIM fluctuations are sufficiently small.The approximation neglects third- and higher-order inverse-FIM terms under that condition.Source: A. PPP-Based Stochastic Satellite Model · p. 5
  • For br = 0, the vertical satellite-assisted FIM converges to a finite value, so sustained vertical information growth depends on terrestrial-network geometry.The resulting overall satellite-assisted FIM grows logarithmically with Dmax as Dmax → ∞ for br = 0.Source: A. PPP-Based Stochastic Satellite Model · p. 6
  • At fixed cooperation radius, root-CRLB scales as O(K̄^-1/2), while enlarging the radius at fixed density yields O((ln K̄)^-1/2) for br ≠ 0.The analysis therefore gives faster asymptotic localization gains from increasing nearby satellite density than from adding increasingly distant satellites.Source: A. PPP-Based Stochastic Satellite Model · p. 6

B. Earth-Curvature-Aware Walker Constellation Model

The Walker model captures Earth curvature, orbital structure, satellite motion, visibility, and changing sensing geometry. It approximates the average hybrid CRLB using visible-cap observations and a Taylor expansion around the mean FIM.

B. Earth-Curvature-Aware Walker Constellation Model

  • The model represents circular Walker orbits at altitude H and accounts for orbital-plane structure, satellite motion, time-varying slant range and line of sight, elevation, and visibility.It neglects Earth rotation over the short observation interval while retaining satellite motion.Source: B. Earth-Curvature-Aware Walker… · p. 6
  • For closed-form moments, visible satellite-time observations are approximated as uniform over the visible spherical cap, with cross-observation covariances from orbital structure and phasing neglected.Each visible satellite-time pair counts as one observation.Source: B. Earth-Curvature-Aware Walker… · p. 7
  • Under the visible-cap approximation and sufficiently small Walker-FIM fluctuations, a Taylor expansion provides a closed-form approximation of the average Walker CRLB.The approximation neglects third- and higher-order terms in the inverse-FIM expansion.Source: B. Earth-Curvature-Aware Walker… · p. 7

IV. CRLB-ORIENTED COOPERATIVE SATELLITE SELECTION

The section develops a CRLB-oriented strategy for selecting cooperative satellites, accounting for both sensing reliability and geometry. It derives marginal-gain bounds and an ordering condition to guide the selection.

  • The proposed greedy strategy sequentially minimises the hybrid trace-CRLB while accounting for sensing reliability and geometry.Source: §IV · p. 7
  • The analysis characterises each candidate’s marginal trace-CRLB reduction and examines directional complementarity and information saturation.Source: §IV · p. 7
  • The section derives analytical bounds and a sufficient ordering condition, then compares satellite-only with hybrid-aware selection.Source: §IV · p. 7

A. CRLB-Oriented Satellite Selection Criteria

The selection criteria define visible Walker satellites as candidates and evaluate subsets using the hybrid trace-CRLB. Because exhaustive search becomes computationally prohibitive with many visible candidates, the paper adopts a low-complexity greedy procedure.

A. CRLB-Oriented Satellite Selection Criteria

1) Marginal CRLB Analysis:

The marginal trace-CRLB reduction from adding a candidate satellite is nonnegative. It is strictly positive whenever the satellite contributes nonzero FIM information.

1) Marginal CRLB Analysis:

2) SCNR Ordering and Geometric Complementarity:

Candidate-satellite gains depend on both information strength and directional complementarity with the current hybrid FIM. Thus, equal-strength candidates with different geometries need not yield the same trace-CRLB reduction.

2) SCNR Ordering and Geometric Complementarity:

3) Bounds and Ordering Conditions for Marginal CRLB Reduction:

The analysis bounds each candidate satellite’s marginal trace-CRLB reduction and gives a sufficient condition for ordering candidates. The resulting greedy procedure selects satellites by exact marginal gain, alongside strongest-SCNR and angular-diversity benchmarks.

3) Bounds and Ordering Conditions for Marginal CRLB Reduction:

  • Eigenvalue bounds on each candidate’s information contribution bound its marginal trace-CRLB reduction and yield a sufficient condition for ordering two satellites.The condition guarantees that one candidate’s reduction exceeds another’s when the stated bound-based ordering holds.Source: 3) Bounds and Ordering Conditions for… · pp. 8–9

4) Greedy Satellite Selection:

  • At each iteration, the CRLB-greedy procedure adds the candidate satellite producing the largest exact marginal trace-CRLB reduction.The selected subset is evaluated using the SCNR-weighted hybrid FIM, accounting jointly for sensing-link reliability and geometry-dependent information diversity.Source: 4) Greedy Satellite Selection · p. 9

5) Benchmark Selection Criteria:

  • The strongest-SCNR benchmark selects satellites with the largest ηs, while angular diversity selects for maximum minimum circular azimuth separation.Angular-diversity selection is initialized with the satellite having the largest aggregate SCNR.Source: 5) Benchmark Selection Criteria · p. 9

B. Satellite-Only and Hybrid-Aware Satellite Selection

Across the PPP and Walker models, the paper validates approximations and examines localization gains from satellite cooperation and geometry. Its CRLB-oriented selection performs near exhaustive search with substantially lower complexity, while hybrid-aware selection helps more under directionally imbalanced terrestrial deployments.

B. Satellite-Only and Hybrid-Aware Satellite Selection

A. Simulation Setup

  • The simulation uses 500 km satellite altitude, a 60 s sensing interval sampled at five instants, and a 10° minimum satellite elevation for visibility.The setup also uses a 15 GHz carrier, four UAV targets, and eight satellites.Source: A. Simulation Setup · p. 9

1) Analytical Validation:

  • In the PPP model, hybrid root-CRLB decreases as the mean number of cooperative satellites grows, while larger cooperation radii yield progressively weaker marginal gains.The percentile region also contracts as more satellites contribute, indicating reduced sensitivity to individual PPP realizations.Source: 1) Analytical Validation · p. 10
  • Under the Walker model, root-CRLB decreases as constellation size and mean visible satellite-time observations increase across the considered approximations and simulations.Exact Walker geometry deviates moderately from the approximation, particularly for sparse constellations; full-SCNR results further deviate without changing the decreasing trend.Source: 1) Analytical Validation · p. 10, Figure 4 · p. 10

2) Geometric Effects:

  • More terrestrial radar receivers and denser Walker constellations reduce root-CRLB, but gains from adding orbital planes or satellites flatten as the constellation becomes dense.The joint constellation-design view shows greater sensitivity to expansion in the sparse region than in the dense region.Source: 2) Geometric Effects · p. 10, Figure 5 · p. 11
  • Higher satellite altitude generally increases root-CRLB, while increasing target height rapidly degrades terrestrial-only accuracy and leaves vertical localization weaker than horizontal.Satellite-assisted sensing is less sensitive to target height because of its high-elevation directions, mitigating the terrestrial-only degradation in the hybrid scheme.Source: 2) Geometric Effects · p. 10, Figure 6 · p. 11

3) Satellite Selection:

  • The CRLB-oriented greedy method achieves the lowest root-CRLB among the compared selection criteria and approaches exhaustive performance with substantially lower search complexity.Exhaustive search requires up to approximately three orders of magnitude more subset evaluations; hybrid-aware selection’s advantage is larger under asymmetric terrestrial deployment.Source: 3) Satellite Selection · p. 10, Figure 7 · p. 11, Figure 8 · p. 11

VI. CONCLUSION

  • The paper combines a PPP-based stochastic model with an Earth-curvature-aware Walker model and validates the analytical approximations using numerical results.The Walker model captures orbit-constrained, time-varying sensing geometry; the selection strategy jointly accounts for sensing reliability and directional information complementarity.Source: §VI · p. 11

APPENDIX A PROOF OF PROPOSITION 1:

The appendices derive the PPP second-order correction and establish asymptotic behavior under fixed-geometry and large-radius conditions. The root-CRLB scales as O(K̄^-1/2), while the vertical FIM is O(1) as Dmax grows when br = 0.

APPENDIX A PROOF OF PROPOSITION 1:

  • Campbell-theorem evaluation builds the second-order correction from Qij, using azimuthal symmetries Qyy = Qxx and Qyz = Qxz.Second-order radial moments are evaluated using I(ν) before substitution into Qij.Source: Appendix A · pp. 11–12

APPENDIX B PROOF OF THEOREM 1

  • The root-CRLB decreases proportionally to 1/√K̄, with the Taylor correction decaying faster than the first-order CRLB.The proof fixes Dmax and H as K̄ grows; the correction satisfies K̄T_taylor = O(K̄^-2).Source: Appendix B · p. 12

APPENDIX C PROOF OF LEMMA 1

  • The horizontal-contribution derivation uses a Dmax ≫ h approximation to simplify a factor involving Dmax²/h².The supplied proof fragments do not state the resulting horizontal scaling law explicitly.Source: Appendix C · p. 12

APPENDIX D PROOF OF LEMMA 2

  • For br = 0, the vertical FIM remains O(1) as Dmax → ∞.The vertical contribution is analyzed in the large-radius regime.Source: Appendix D · p. 12

APPENDIX E PROOF OF PROPOSITION 2:

The Walker-model proofs obtain a tractable hybrid-FIM approximation using horizontally isotropic visible observations. The second-order Taylor correction is evaluated under an approximation that neglects cross-observation covariances.

APPENDIX E PROOF OF PROPOSITION 2:

  • The Walker-model approximation assumes large-system, horizontally isotropic visible observations when averaging the satellite-assisted FIM over local horizontal azimuth.The visible-observation LoS unit vectors are represented under this approximation.Source: Appendix E · pp. 12–13
  • Effective weighted Walker moments are approximated as Im ≈ Mvisμm for m = 0, 1, 2.These moment approximations are substituted into the averaged-FIM expressions.Source: Appendix E · p. 13

APPENDIX F PROOF OF PROPOSITION 3:

  • The second-order Taylor correction uses weighted visible moments and neglects cross-observation covariances when evaluating the Walker-FIM fluctuation moments.The proof defines diagonal and off-diagonal second-order moments under this approximation.Source: Appendix F · p. 13
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