Measurement Simulation

ACONS synthesises observables by combining satellite/user ephemerides, link budgets, and tracking loop noise models. This page documents the signal assumptions and equations implemented in src/measurement_simulator.py, which writes the full measurement catalogue to measurements.json.

Geometry

For a satellite with position \(\mathbf{s}\) and velocity \(\mathbf{v}_s\), and a user with position \(\mathbf{u}\) and velocity \(\mathbf{v}_u\), the simulator evaluates:

\[ \rho = \|\mathbf{s}-\mathbf{u}\|, \qquad \dot\rho = (\mathbf{v}_s-\mathbf{v}_u)^\top\frac{\mathbf{s}-\mathbf{u}}{\rho}, \]

the one-way line-of-sight range and range rate expressed in metres and metres-per-second. These quantities provide the geometric core for the range and Doppler observables stored in measurements.json (fields true_range_m, range_rate_true_mps).

Signal model

The link budget follows a Friis formulation. For each time sample the simulator evaluates

\[ \frac{C}{N_0} = \frac{\mathrm{EIRP}\cdot L_p \cdot g_r}{k_B\,T_{\mathrm{eq}}}, \]

where \(\mathrm{EIRP}\) is the transmit pattern evaluated at the user look angle, \(L_p\) is the free-space path loss, \(g_r\) is the receive antenna gain, \(k_B\) is Boltzmann’s constant, and \(T_\mathrm{eq}\) is the equivalent noise temperature computed with Friis’ cascade (antenna plus LNA). measurement.transmitter and measurement.receiver_rf control the gain patterns and RF parameters. The resulting \(C/N_0\) (field cn0_dbhz) drives both measurement availability and the thermal noise models described below.

Range and Doppler observables

Measured pseudoranges can be written as

\[ \hat{\rho}_u = \|\tilde{\mathbf{r}}_s - \mathbf{r}_u\| + c\,(t_u - \tilde{t}_s) + \varepsilon, \qquad \hat{\dot{\rho}}_u = \|\tilde{\dot{\mathbf{r}}}_s - \dot{\mathbf{r}}_u\| + c\,(\dot{t}_u - \tilde{\dot{t}}_s) + \dot{\varepsilon}, \]

where \(\tilde{\cdot}\) denotes the transmitted (clock-biased) satellite quantities and \(t_u\), \(\dot{t}_u\) represent the user clock bias and drift. The simulator realises these observables via

\[ \rho = \|\mathbf{r}_s - \mathbf{r}_u\| + \Delta t_u + \Delta t_{sv} + \varepsilon_{DLL}, $$ $$ \dot{\rho} = \|\dot{\mathbf{r}}_s - \dot{\mathbf{r}}_u\| + \Delta\dot{t}_u + \Delta\dot{t}_{sv} + \varepsilon_{FLL}, \]

with the following components:

• \(\Delta t_u = b_u(t)\) is the user clock bias in metres. It is obtained by integrating the drift series generated from the Allan deviation curve configured under measurement.oscillator.
• \(\Delta \dot{t}_u = \dot{b}_u\) is the corresponding clock drift (metres-per-second) converted from the OCXO fractional-frequency noise and applied uniformly across all satellites at a given epoch.
• \(\Delta t_{sv}\) and \(\Delta\dot{t}_{sv}\) model satellite clock/orbit errors (“SISE”). The scenario provides ODTS one-sigma scalars (metres and metres-per-second) under measurement.sise. These sigmas are already defined along the line of sight, so the simulator applies a zero-mean Gaussian draw with that standard deviation to every satellite at a given epoch. Clock bias/drift SISE use the corresponding scalar values provided in the same block and are added on top of the orbit term for one-way observables. The simulator stores the realised series as sise_error_m (combined term for range-like observables), sise_orbit_error_m, sise_clock_bias_error_m, sise_orbit_rate_error_mps, and sat_clock_drift_error_mps. The per-epoch variances are exported alongside the measurements as sise_range_variance_m2, sise_two_way_range_variance_m2, sise_range_rate_variance_mps2, and sise_two_way_range_rate_variance_mps2, enabling the EKF to enlarge its innovation covariance with the same SISE model used by the simulator.

• \(\varepsilon_{DLL}\) and \(\varepsilon_{FLL}\) are the thermal tracking errors derived from the DLL/FLL models detailed below.

• The simulator expects a single measurement.oscillator entry with Allan deviation samples at two or more averaging times. Values can be supplied at arbitrary \(\tau\) and the ingestion stage interpolates across them, ensuring the truth clock matches the specified profile. The measurement.sise block further refines the satellite-side position/velocity and clock errors using the ODTS sigmas, for example:

measurement:
  sise:
    position_sigma_m: 2.89
    velocity_sigma_mps: 7.5e-4
    clock_sigma_m: 7.5
    clock_drift_sigma_mps: 7.5e-4

These values drive both the simulated observables and the innovation covariance used by the EKF.

Two-way measurements remove the clock terms while retaining the same thermal noise. The simulator therefore uses

\[ \rho^{2w} = \|\mathbf{r}_s - \mathbf{r}_u\| + \Delta t_{sv,p} + c\,t_{cb} + \varepsilon_{DLL}, \qquad \dot{\rho}^{2w} = \|\dot{\mathbf{r}}_s - \dot{\mathbf{r}}_u\| + \Delta\dot{t}_{sv,p} + \varepsilon_{FLL}, \]

where \(t_{cb}\) is the configurable two-way calibration bias (0.5 ns → 0.15 m by default) and \(\Delta t_{sv,p}\), \(\Delta\dot{t}_{sv,p}\) reuse the ODTS orbit and velocity draws described above, which remain present even after the clock terms cancel. measurements.json records the realised user and satellite clock terms (user_clock_bias_m, user_clock_drift_mps, sise_error_m, sise_orbit_error_m, sise_clock_bias_error_m, sise_range_variance_m2, sise_two_way_range_variance_m2, sise_orbit_rate_error_mps, sise_range_rate_variance_mps2, sise_two_way_range_rate_variance_mps2, sat_clock_drift_error_mps, etc.) alongside the observable values, allowing downstream tools to inspect the individual contributors.

Deep-space assets typically limit uplink time, so the simulator constrains the two-way observables to short windows. measurement.two_way_availability_minutes specifies how long each contact lasts, while measurement.two_way_availability_cadence_minutes defines how often the windows repeat (both default to 60, yielding continuous availability). When the cadence exceeds the duration the simulator gaps out the two-way range and range-rate columns between contacts, ensuring the EKF only ingests the available passes. Setting the duration to 0 removes all two-way data entirely.

measurement:
  types: [range, range_rate, two_way_range]
  two_way_availability_minutes: 10     # 10-minute windows
  two_way_availability_cadence_minutes: 30  # repeating every 30 minutes
  two_way_selection_strategy: window_locked  # lock to the highest-C/N0 link at window start

measurement.two_way_selection_strategy controls whether the highest-\(C/N_0\) satellite is chosen at every epoch inside a contact (per_epoch, legacy behaviour) or only once when the window opens (window_locked, keeping the same spacecraft throughout the window). The latter better emulates ground stations that avoid reconfiguring beams multiple times within the same pass.

Thermal noise and tracking loops

The measurement block in the scenario file exposes the delay-lock (DLL) and frequency-lock (FLL) loop parameters. The simulator uses the Methodology jitter model to convert \(C/N_0\) into measurement noise:

\[ \sigma_\text{DLL} = \frac{c/f_\text{code}}{2\,\Delta} \sqrt{\frac{B_L}{C/N_0}\left(1+\frac{1}{T_i\,C/N_0}\right)}, \]

\[ \sigma_\text{FLL} = \frac{\lambda}{2T_i} \sqrt{F\frac{B_L}{C/N_0}+\frac{1}{T_i\,(C/N_0)^2}}, \]

where

• \(B_L\) — loop bandwidth (0.5 Hz by default),
• \(T_i\) — coherent integration time (20 ms),
• \(\Delta\) — early–late spacing (1 chip),
• \(\lambda = c/f_c\) — carrier wavelength,
• \(F\) — PLL/FLL scaling factor (1 above 35 dB-Hz, 2 below, configurable).

The resulting \(\sigma_\text{DLL}\) (metres) and \(\sigma_\text{FLL}\) (converted to m/s) populate the range_noise_std_m and range_rate_noise_std_mps fields. The EKF reads these values directly from measurements.json.

Measurement diagnostics

measurement_simulator.py retains the full set of metadata per observation (satellite/user position and velocity, LOS vector, azimuth/elevation, visibility flag). The reporting layer applies a robust Median Absolute Deviation (MAD) filter before generating summaries and plots, ensuring extreme outliers do not skew the diagnostics. See Outputs & Reporting for the derived CSV/plot artefacts.

Trace logging

Set --log-level TRACE when running acons simulate … to include exemplar measurement breakdowns in simulate/simulate.log. The simulator logs one representative one-way and two-way range sample plus the corresponding range-rate observables, highlighting the geometric term, user clock bias or drift, satellite orbit/clock errors, calibration bias, and DLL/FLL thermal noise that sum to each observation.

Implementation notes

The simulator code mirrors the structure described above. Geometry extraction, link-budget evaluation, and observable synthesis live in dedicated helpers (_satellite_geometry, _link_budget_metrics, _generate_range_observables, _generate_rate_observables), while simulate_measurements orchestrates the per-satellite loop. The split keeps the numeric models focused and makes it easier to extend the simulator with alternative antennas, additional measurement types, or unit tests that exercise a single stage of the pipeline.