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title	summary	tags
Ephemeris sampling and interpolation	When to sample a track once and query it many times, and what the interpolation costs in accuracy.	ephemeris, interpolation, sampling, hermite

Ephemeris sampling and interpolation

A coverage search needs the position of a satellite at many instants. Asking the propagator for each one is correct and slow. Sampling the track once and interpolating between the samples is fast and introduces a small, bounded error. This document gives the trade.

Definition

An ephemeris is a table of satellite states at a sequence of instants. A sample holds the time, the position, and the velocity at one instant.

Interpolation evaluates the state between two samples without calling the propagator, using the two samples that bracket the requested instant.

Mathematical meaning

The interpolation is a cubic Hermite polynomial per component. It uses the position at both endpoints and the velocity at both endpoints, so it matches the value and the first derivative at the ends of every interval:


p(s) = h_{00}(s) \, p_0 + h_{10}(s) \, \Delta t \, v_0 + h_{01}(s) \, p_1 + h_{11}(s) \, \Delta t \, v_1

where s is the fraction along the interval in [0, 1], dt is the interval length, and


\begin{aligned}
h_{00} &=  2 s^3 - 3 s^2 + 1 \\
h_{10} &=      s^3 - 2 s^2 + s \\
h_{01} &= -2 s^3 + 3 s^2 \\
h_{11} &=      s^3 - s^2
\end{aligned}

are the Hermite basis polynomials. The velocity is the derivative of the same polynomial with respect to time.

Because the velocity is used at both ends, the interpolation is C^1: the position and the velocity are continuous across a sample boundary. A table built from position alone would have a discontinuous velocity at every sample, which is visible in a plot as a kink and is wrong for anything that differentiates the track.

The interpolation error over one interval scales as the fourth power of the interval length times the fourth derivative of the position. That is the quantity that sets the sample spacing.

Unit convention

Sample times are in seconds from whatever origin the caller chose. A sampler defined against a propagator usually takes its times in the propagator's own unit, minutes from an element epoch, and converts to seconds for the table.
Position is in kilometers and velocity in kilometers per second, the units of the propagator that produced the table.

Applicability

The interpolation is valid inside the interval the table covers. Outside it, the polynomial extrapolates and the error grows quickly; a query outside the covered range returns no value unless the caller explicitly asks for the nearest sample.

The error is bounded by the sample spacing. A useful rule for a low Earth orbit, where the period is about 90 minutes:

Sample spacing	Interpolation error
60 s	Below a meter
300 s	A few meters
600 s	Tens of meters
1800 s	Hundreds of meters

These values are for a smooth low Earth orbit. An orbit with a high eccentricity, where the angular rate changes sharply at perigee, needs a shorter spacing near perigee or a uniform spacing set by the perigee rate.

The table also stores the velocity, so it can answer a query for the velocity as well as the position. A table built from positions alone cannot, and would need a finite difference, which is less accurate.

Comparison of query strategies

Strategy	Cost per query	Accuracy	Memory	When it applies
Propagate on demand	One propagation	The model's own	None	A few thousand queries
Uniform table with Hermite interpolation	Tens of operations	Set by the spacing	One record per sample	Many queries over a known interval
Non-uniform table with tight spacing at perigee	Tens of operations	Set by the local spacing	More samples near perigee	A high eccentricity orbit
Chebyshev polynomial per arc	Tens of operations	Set by the degree and the arc	A coefficient set per arc	A long smooth arc
Two-body propagation from osculating elements	Very low	Degrades fast because the model is incomplete	None	A rough check only

Chebyshev approximation is used by precise ephemeris products and offers a tighter error for a given number of coefficients than a uniform table. It is inconvenient for a table built directly from a propagator, because the coefficients are fitted to the arc rather than read off it, which is why a uniform table is the usual form for a table derived from propagation.

Source

Cubic Hermite interpolation is the standard two point, two derivative polynomial; the basis above is the usual form.
The use of an interpolated table for access and coverage searches is the standard approach in mission analysis, as described in Vallado, Fundamentals of Astrodynamics and Applications.

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