Interpolation Methods¶

SP3 and CLK products sample at 15-minute or 5-minute intervals. RINEX observations arrive at 1–30 s. Interpolation bridges the gap — producing per-epoch satellite positions and clock corrections at the exact observation timestamps.

Hermite Cubic Splines

Used for SP3 ephemerides (satellite XYZ positions).

SP3 files include both positions and velocities, enabling Hermite interpolation — C¹ continuous, sub-millimetre accuracy for IGS final products.
Piecewise Linear

Used for CLK clock corrections.

Clock files provide no derivative information and exhibit discontinuities at manoeuvre events. Linear segments between knot points give sub-nanosecond accuracy without over-fitting.

Hermite Cubic Splines (Ephemerides)¶

Physical motivation: orbital motion is smooth — well-suited to polynomial interpolation. Positions and velocities from SP3 files determine the Hermite polynomial coefficients uniquely over each 15-minute interval.

from canvod.auxiliary.interpolation import Sp3Config, Sp3InterpolationStrategy

config = Sp3Config(
    use_velocities=True,          # use velocity columns from SP3 (recommended)
    fallback_method="linear",     # fall back if velocity missing
    extrapolation_method="nearest",
)

interpolator = Sp3InterpolationStrategy(config=config)
result = interpolator.interpolate(sp3_data, target_epochs)

Parameter	Default	Effect
`use_velocities`	`True`	Enables Hermite mode (higher accuracy)
`fallback_method`	`"linear"`	Used when velocity columns absent
`extrapolation_method`	`"nearest"`	Behaviour outside the SP3 time span

Accuracy

Hermite splines achieve < 1 mm position error at 30 s cadence for IGS final products. Rapid products are typically < 5 mm.

Piecewise Linear Interpolation (Clock Corrections)¶

Clock corrections are not smooth — receiver and satellite clock models are periodically updated, causing step discontinuities. Linear interpolation between adjacent CLK knots avoids fitting across those jumps.

from canvod.auxiliary.interpolation import ClockConfig, ClockInterpolationStrategy

config = ClockConfig(
    window_size=9,          # number of knots to fit (odd, centred)
    jump_threshold=1e-6,    # discontinuity detection threshold (seconds)
    extrapolation="nearest",
)

interpolator = ClockInterpolationStrategy(config=config)
result = interpolator.interpolate(clk_data, target_epochs)

Jump detection

When |Δt| > jump_threshold between consecutive CLK knots, the interpolator treats the gap as a boundary and never interpolates across it. The nearest valid knot value is used instead.

Custom Strategies¶

Implement InterpolationStrategy to plug in custom interpolation logic:

from canvod.auxiliary.interpolation import InterpolationStrategy
import xarray as xr

class CustomStrategy(InterpolationStrategy):
    """Example: Lagrange polynomial interpolation."""

    def interpolate(
        self,
        aux_ds: xr.Dataset,
        target_epochs: xr.DataArray,
    ) -> xr.Dataset:
        # Your implementation here
        return interpolated_ds

The strategy is injected into the auxiliary pipeline — no other changes required.

Accuracy Summary¶

Data	Method	Expected accuracy	Notes
SP3 final (IGS)	Hermite	< 1 mm	Requires velocity columns
SP3 rapid (IGS)	Hermite	< 5 mm	Typically available within 17 h
SP3 ultra-rapid	Hermite	few cm	Predicted half
CLK final	Linear	< 0.1 ns	Sub-centimetre equivalent
CLK rapid	Linear	~0.5 ns	Typically available within 17 h